Help me figure this out, please.

[Pookey]

Hi all,
I'm hoping you can help me trouble-shoot what's wrong with my truck. Full disclosure: I know nothing about engines but want to learn. Figure I can try to address some simple things myself before resorting to a mechanic.

1962 Chevy stepside
1963 327 engine (from a corvette is what I'm told)

I recently moved cross country and had the truck auto-transported. Drove the truck for a little over an hour to get it home (this was 2 weeks after it was delivered to my friend). During that trip I felt it shimmy when it hit 45 mph. That would fade once I got to around 55mph.

After about 3-4 weeks of sitting in the garage, I cranked it up. Very hard start, very rough idle then eventually it would die. I had pulled it out of the garage and it was a good while (hours) before I could get it to start again to get it back in the garage. It was sitting at an incline in the driveway, if that matters.

It was running great before the move. Also, aside from the engine issue, the driver's side is sitting lower than the passenger side. That's new, too. I'm more concerned right now with the engine issue.

Any ideas would be greatly appreciated. I've tried to find info on this and so far I have the following:

spark plugs, fuel filter, carb filter, fuel pressure regulator, ignition coil or module, fuel pump.

I did spray a can of carb cleaner when it was running but that didn't help it.

Thanks for any and all help.