I can't figure out why my Ford won't run!

A few weeks ago I bought a 69 Falcon. Looks great engine seemed to be in good shape.
Ran great for the first week and then slowly declined. I gave it a tune-up (plugs, wires, cap, rotor, set points, air, oil and fuel filters) and that really seemed to help a lot. Then I took a 100 mile trip to the beach in it and towards the end of the trip it started choking up and would barely run. I could idle, or keep a constant speed just fine, but I couldn't accelerate. I thought it was a carb.
I had to have it towed home because it just got to the point where it wouldn't even take off. it would just sputter, backfire and die. I took off the carb and had it professionally rebuilt by a reputable shop. I put the carb back on and the car ran better than ever.... For about a week. Now it's back to the same problem. It'll idle forever (rough) but I can't get the thing moving at all.
Thought for a minute it could be a vacuum leak somewhere but I just replaced all the lines and made sure the carb was down nice and tight. That changed nothing.

I'm not sure what else to check. I'd love some suggestions on where to go from here.

Thanks yall!

I'm not really sure what else to check out.

Check for plugged fuel filter. Old car probably picked up debris from tank. Check fuel pump pressure. Lastly put a timming light on it to see if the mechanical advance is stuck.

Took the filter off and cranked it over. Gas pumps out through the filter with authority.
And from what I understand it wouldn't be the timing because there have been a few times since I got it that it's run just perfect.

I've yet to change the condenser. Could this be the problem? My mechanic friend next door suggested that I replace my coil too.

The condensor may be the problem. pull the cap and observe the points to see if there are sparks jumping the gap while someone else cranks it over for you. The condensor stores the unwanted power between point openings. While you're at it check the mechanical weights for sticking. It is possable that the coil is breaking down, but it would need to be tested with a spark tester.

The last 2 posts were spot-on.
The points and condensor rarely last more than 10,000 miles and are cheap to buy. For the effort involved in setting the points up, you may as well change them and the condensor together.

Be sure to use a dwell tachometer to do the job right. And buy TWO sets of points-and-condensors. Keep the second set in the glovebox for emergencies.

Check the coil to make sure it is not hooked up backwards. The negative (-) terminal must be hooked up th the wire that goes to the distributor. The positive terminal (+) must be hooked up to a positive electrical source. Note, this positive wire should show 12 volts when the engine is cranking, but only 9 volts when the engine is running.

If the hook-up is good, replace the coil. Faulty coils will behave as you describe when they warm up as the engine runs. Don't get a fancy brand, just use a standard old-style generic black coil. If your auto-parts place does not have them, try a farm tractor supply store.

BTW, original stock Ford coils of that era tend to be pretty short. You will find a new old-style generic coil will be longer than the original, but will fit in the original bracket just fine. The longer coils have more windings than the original Ford pieces and produce a stronger spark. However, they will have the identical electrical hook ups.

Well I pulled the coil out of my Gremlin and threw that on.
That definitely made a big difference. It idles a lot smoother and accelerates just fine.
I replaced the condensor as well. That didn't seem to effect anything.

It still idles pretty rough though. I've definitely had it running better before.

What else do yall think I can check out?

A rough idle can be caused by a retarded ignition timing and/or a vacuum leak and/or improper carb adjustment . Even though its been rebuilt, the idle mixture screws and idle speed may need some fine-tuning.

If you're not concerned about originality, converting to an electronic ignition is fairly inexpensive, and much more reliable than a points and condenser system.