3&6 missing, stalling when put in gear

My valiant has been a garage queen for 5 months because of this. '68 Valiant with a 225 and auto trans. It was my daily driver for a long time. It's got a reoccurring issue that comes and goes.
First off, some things about my car.
-No choke. I removed it years ago because the mechanism was missing/broken when I first got it. I live in a tropical climate, and usually 3 pumps on the gas pedal starts it on the first crank.
-Remanned carb. It has had this same issue with the original and remanned carb, but also run right on both carbs.
-remanned distributor from a '66 slant six. Found this out while shopping for a new set of points all day. Car ran good when I originally got this dizzy.
-fuel pump and filter are new. This car has always had a plastic fuel cell in the trunk, and it uses an inline clear plastic filter.
-no vacuum leaks I can find. Not on the throttle shaft, base gasket, intake manifold, pcv valve. I've tried capping the vacuum advance and connecting it, no difference.
-Compression reads between 130-140 psi on all cylinders.

Recently I've replaced the cap, rotor, wires, condenser, timing chain and gears and associated gaskets. Timing runs 5* btdc, spot on . I've checked the timing, run it with and without the vacuum advance, checked points gap, and the valves were adjusted about 3 months before it got parked due to the issue it's having.

THE ISSUES:
On cold starts recently, it runs rough, sounding like 4-5 cylinders. If I let it idle for a while, it will eventually smooth out a little. Put it in drive or reverse and it will stall almost immediately when touching the throttle. If I feather the gas, I can get it moving and it becomes much more willing to move once it's rolling. I've checked the accelerator pump, and it puts a nice strong stream into the carb. Carb is spotless inside. Once warmed up and moving, it's got a very occasional cough at cruising speed, but not enough to notice unless you're listening for it.
I throw the timing light on each plug wire, #3 has an occasional pause in spark over 2000 rpms, #6 is firing about half the time. Strange thing is, #6 has ALWAYS run rich as long as I've owned the car. Running good and peppy, or shitty and rough, #6 plug is ALWAYS black.... with multiple tuneups, 2 different distributors, electronic and points. #6 is black.

Anybody got a clue for me what to check next?

May be time to pull the valve cover and see if you have good movement on all the rockers. Especially #6, could have worn lobe,rocker or maybe a broke valve spring?

Ok, I'll check em out this afternoon.

Since 3 and six are sequential in the firing order the possibility of the exhaust not opening (or opening very little) on 3 and polluting the intake charge on 6 could explain this.

One possibility for #6 is a fuel leak in the carb; excess fuel could be spilling over into the plenum and will tend to run downhill towards #6. Look down in the carb while idling; if there is fuel dribbling in actual drops into the carb, then it is not right.

If you have changed plugs and wires at #6 and that did not fix it, that seems to rule them out. I assume you have changed the distributor cap?

Have you checked compression on all cylinders?

Cylinder balance test.
At slow idle pull one wire off at a time, and note rpm drop.Be accurate,write it down.Betcha #3 is nearly zero rpm drop. If true, check the cam action,as already mentioned.Engine could still make the compression number with a bad exhaust lobe.
With #6 wire disconnected, and rpm drop noted, leave wire off, and disconnect #3 as well.Record any change.

BTW; adjusting valves on a slanty requires a narrow feeler gauge. The valve stem tends to pound groves in the rocker arms, requiring narrow feelers. If you used the standard wide ones, they will straddle the groove and give you a false reading.The clearance must be checked with the lifter on the base circle.Sometimes the base circle isn't round anymore.

You may want to replace the float in the carb if you have not already. They tend to get heavy over time & they're cheap preventive maintenance.

Sorry it's been a while, I was busy getting married
I checked compression on all cylinders, and they all clock in over 135psi. Checked for vacuum leaks, no dice. I plugged the vacuum from the brake booster, and there wasn't any change in idle. The float is less than a year old, so I dunno if it's much of an issue. All the valves seem to move normally, and valve gap is set. They were slightly off, but there was no change with an adjustment. I don't have narrow feelers, so I'm not sure if a groove in the rockers was an issue, but they don't look worn.

I'm going to see if my buddy has a dwell meter and make sure that's all peachy, but the cap, rotor, wires, points, distributor, timing chain and gears are all new within the last year (and within 6 months from the last time I actually drove the damn thing)

I'm at a loss. I have a spare head head and two barrel manifold I wanted to throw on it after a complete rebuild on the head, but I'm hesitant to move forward with that and still run into the same issue afterwords, as well as any new issues I've added by swapping all that stuff.

This is not a dwell issue. This is an issue with 3 and 6 only. Dwell affects all 6.
Here's what I would do, since you don't want to do the cylinder balance test. I would swap the wire from #6 to another cylinder, and see if the problem follows the wire. I would do the same for #3. If no change, I would swap the plugs to different holes, also to see if the problem follows the plugs. You see, the timing lite is telling you that those two cylinders are not firing correctly. Whether its a plug or a wire or faulty A/F charge, you have to find it by process of elimination. The easiest to prove is the ignition system.
So if swapping the plugs and wires around makes no difference, then you have proven the ignition system is not the source of the issue.
Next would be the fuel system.Since it runs on the other cylinders, just fine, the carb is fine( it cannot tell one cylinder from another); so that just leaves the intake manifold and the connection of it to the head. Since you say you cannot find a vacuum leak,you have sorta proven that system is ok.
That just leaves a mechanical issue. But you have proven the compression is ok and the valve clearance is, at the very least, not too tight. So, what you are saying is the engine is junk.
-But hey, there are still several things to prove. #1 is valve-action; both valves in both questionable cylinders must be proven to be fully opening, and fully closing, and doing it at the right time, and reliably. You see, full compression does not prove that the cam lobes are all there or that the valve springs are not broken or that they have sufficient pressure to do their jobs. So that would be the next place to look.

But back up the busaminit.
You say you could not find a vacuum leak. I just want to throw this out there.Suppose the intake was cracked in the area between it and the exhaust manifold,where they are bolted together? If the engine was sucking exhaust there it would run kinda crappy.And you wouldn't find it in the normal way of checking. Whether or not it would affect just two cylinders, I cannot say. Nor can I figure out how or why the timing lite would see it as it now does. So this would be just something to keep in the back of your mind, if you cannot find the issue in the valve-action.

I also find it perplexing, that the only thing that 3 and 6 have in common is that they fire one after the other, leading a guy straight to the distributor cap; and that the cap is new.Hmmmmmmmmmmmm

I'm going to try the balance test next, but ran out of time this morning... was scheduled to start work at noon... just enough time to get started but not finish a project
I'm going to throw another wrench into the proverbial works here and mention that this issue has happened before. Same symptoms, ect. However, at the time I had gathered a bunch of new parts to put on; oil pump, intake/exhaust gasket, valve cover gasket, oil pan gasket, coil, cap, rotor, plugs, wires, points... damn near everything I could replace in a weekend since it had a bunch of old bits that came with the engine when I bought it. It went away. Now it's back, a little at first and getting progressively worse. Last time everything I did seemed to make it a *little* better, but no one thing seemed to solve the issue alone. I'm gonna set aside this weekend to work on it with another wrenching buddy over and see if we can figure it out.