69 Cougar won't stay running (351W)

I’m hoping someone can help me out here, as I’m having trouble with a 69 base model (351W) I recently bought. The car was a garage find (sat for 15 years), and after draining the old fuel out, ran fine for the first week or so. After stalling on me last week, I noticed the fuel coming to the pump was off color, obviously crud on the inside of the tank had worked its way loose. Since then, the car will not stay running, is incredibly difficult to start, and idles rough, often backfiring. I’ve done the following, all with no luck:

-Ran clean gas direct from jerry can to fuel pump.
-Replaced points and condenser, set and checked points gap
-Cleaned and rebuilt carb with all new gaskets, float, needles and seats (autolite 2100)
-New fuel filter
-New spark plugs

Anyone know where to go from here? Every so often it’ll straighten out and run smooth for a minute or so before stalling, so I can’t imagine its an internal problem. Any advice is appreciated.

When my car suddenly started to miss & backfire at even speed/deceleration and nearly stall during acceleration, my mechanic suspected a firing problem and did the same as you with points, condenser and plugs while coil, wires and distributor cap all looked OK. This lead to a big improvement but the engine was still missing at around 40 mph. So he looked closer at the distribuor and found that it was kind of “sitting a little loose” in the engine (due to worn gear and/or shaft) causing malfunction of the vaccum advance.
Symptoms of your problem are stronger than mine (engine stalls quickly at idle, no need to accelerate) but the cause seems similar i.e. firing malfunction.

I agree take a look at the distributor vacume advance plate it could be hanging up messing up the timming, also the distributor cap for cracks. I know you changed the condensor but I have had one cause the same problem as the car ran longer the idle would get rough, then start to stall then crank like it had a week battery.
The quality of alot of the new parts is not that good.
Check the oil and make sure ithe fuel pump is not dumping oil into the crankcase years ago I bought 9F91M500030 as a non runner the seller had disconected the starter so the car would not run due to gas in the oil his mechanic had no idea what was going on. Changed the oil and pump and it ran great.
hope the info helps

Turned out to be a bad condenser. Put the 15 year old one back in and it fired right up and ran like a dream.