Sounds like every other CJ out there
Those carbs are 28 years old now. They have their share of issues, but over all they work ok. For the price of a rebuild kit (probably $15-$30) and a night of your time you can take that one apart and rebuild it yourself and probably fix the problem. They really are not bad to work on and that may work great with some attention.
As mentioned before the MotorCraft 2100 carb is also an easy upgrade/replacement. Basically you find an old Ford carb, modify it, adapt it and it works really well on those engines. Not a real expensive conversion either, maybe around $100 when you are all said and done.
Yet another option is a new Weber carb that is meant for the engine. They are hundreds of dollars and probalby work about as well as the MC2100. You pay more money, and get more bling, but to me there is no value gained.
I just feel bad when I hear of guys going to a carb shop and being charged $350 to have their stock carb rebuilt when there are so many other options that are better or far cheaper avalible.