Given what you are saying it is doing, I personally would stop your periodic starting and running of the engine until after you rebuild the carb.
In all likelihood you have some clogged jets as a result of sitting and having deposits building up in the carburetor bowl.
There are a few other things it could be. You could have a very clogged fuel filter, you might change that as a first step to see if that clears things up and would need to be done in any case. Another area that could be the root of the problem is the fuel pump may be failing. You might check to see if your oil is getting contaminated by gasoline (though this can happen from running very rich under the choke) by a torn/failed diaphragm allowing fuel to go directly into your crankcase. I doubt this is the case but it is something to check. You might also very carefully ensure the pump is actually pumping properly after the car has been shown to start by pulling the supply line to the carb and turning the engine over with the end in a plastic cup. It is low pressure but a decent volume so be careful.
When running on the choke it is likely running quite rich at idle and that also applies to running at higher rpm. Once the choke goes off and you try running at something greater than idle it may be running lean when off the choke due to a clogged jet reducing the fuel available.
Rebuilding a carb is not a difficult task in general but it does require you keep things organized and be scrupulous about being clean.
I find that keeping the body of the carb in the same orientation to you, working in a shallow cooking pan so parts can’t roll away and taking a cardboard box and poking small holes in it to hold the parts as you remove them Is a good work process that will aid you in keeping track of things and the order of assembly.
As you are taking jets, emulsion tubes and so on off that you write down the tiny numbers and letters on each part down and where it came from. The numbers/letters can be hard to see but they are there.
Print out the exploded view of the carb and have it close at hand when you start taking it apart.
I would use some carb clean and a brush to clean the exterior of the carb before taking it apart as they can get quite grimy and nasty. You don’t want any of that getting into the carb or being a part of the detail cleaning process.
Get both spray carb cleaner to go through the various passages and a gallon of dip to place the larger sections into which you will then use the spray clean to clear passage ways. Ensure the gallon has a dipping cage or use a metal screen colander to hold the small parts in the gallon dip bucket.
The other Carl is much smarter than I and has lots of experience in rebuilding carbs and will be along shortly to correct the misconceptions I am spouting...
(Damn, there are too many emojis to choose from...)