Back on the road!!!..... for about 5 min (engine help please)

Sorry if this post is a little long, I just want to be as specific as possible.

Alright, 66 Dodge Dart 270, 225 cu, 3sp a/t.... Was running fine all summer then it was stalling out and generally running like crap, sometimes wouldn't start...

Previous issues I was experiencing: Ran fine in P, R, and N.. In D it had shoddy/sporadic/hesitant acceleration and eventually got to the point where it stalled if I wasn't constantly giving it gas.. (at stop lights I would have to keep revving in N, then switch gears and then give it gas soon as I switched gears)... Sometimes it would go ahead and stall as I was braking to slow car. Would even stall in 35 mph traffic if I wasn't at least giving it some acceleration. I gave up on trying to troubleshoot it myself and took it to mechanic. He told me I need carb rebuilt. He ordered Holly kit and did that and he told me it was still stalling so upon further inspection also had distributor changed because he said it was shot as well.

I got my car back and it was hesitant to start. Finally cranked over and was running great. Then when I stopped at a light it seemed fine, then when light turned green it hesitated a little and stalled when I gave it some gas. :( This time when I went to turn key I noticed it really was slow cranking. Got it home and noticed battery was from 2005 and so I bought a new one and BAM fired up with a vengeance and I thought everything was peachy!

I then get 3 miles down the road in traffic and it stalls!!!!! So here are the current issues I am now having: Car starts up fine, idles fine, will stall in reverse if engine is cold, and in Drive it'll start to roll fine and I feel like it could roll forever, but then I give it gas and there is a split-second where it stutters and I feel like I need to give it gas, but feel like if I give it too much gas it will stall out. Additionally, after car has been running its playing the stalling game again when slowing down. Not all the time, but definitely enough where I don't feel safe driving it in traffic.

Very frustrated. Any ideas? I'm a newb to all of this. I'm don't know for sure but I feel like there might be a fuel/delivery problem? Totally speculation. Ayy help from the masters on the forum would be greatly appreciated. Thank you so much.