360 Engine issues

I have a stock late 70’s 360/727 in my 67 Barracuda that has a couple problems. I think I have a solution but would like to run it by you
First: Engine vibrates at idle.
When I say vibrate it’s something I can feel it in the steering wheel and kinda see it in the engine bay. Seems to fade away when above 2000rpm. Motor and tranny mounts are new and tight. Looking at converter I don’t see a weight welded to it anywhere. Also the harmonic balancer Is starting to spit out the rubber.

Would installing a B&M flexplate and replacing the balancer potentially solve the vibration?

Second: Oil pressure at startup after it sits for a day.
Turn the key, engine cranks and starts and for 3-5 seconds there’s no oil pressure. Then you can hear and feel the engine load up. Oil pressure climbs up to 65lbs and the engine quites down. Engine has 65psi cold and 35 hot. Stock oil pan with 6qts of 10w-30 and wix 51515R oil filter.

Guessing the oil is draining back into the pan through the oil pump? Should the pump hold oil on the discharge side? If the oil pump needs replaced will the pan clear the cross member?

Thanks for any input
Jim

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