I'm voting with George on this, Since your engine runs ok running down the road but burps when running at slower speed it most likely some type of vapor lock problem especially if you are forced to use this hippy 10% ethanol fuel.
There are a couple places to check, First check the flex hose line from the firewall to pump, these are notorious for deteriorating and cracking. If they crack even slightly the fuel pump sucks air and there is an issue, if you have not replaced that line recently it might be time for a new line or at least a check.
Second review the entire fuel line system from the carb to the tank for any area that comes near an exhaust manifold or muffler. Radiant heat from these is sufficient enough to cause the fuel to boil within the fuel line and shutting off the fuel flow. While the stock exhaust is on the passenger side and out of the way you still should check how close the feed line is to the driver side exhaust manifold , if you have added dual exhaust this might be the area that needs a look.
The ability to start your car after a short period of cool down time points to some type of fuel vaporization issue. A quick check on your fuel supply is to splice a small clear fuel filter in the flex line from the line to the carb. These are available at the hardware store for lawn movers. This will indicate if you have fuel to the carb or show bubbles building up if you have a fuel vaporization problem, they will also prevent any fine rust from filtering thru the system as a bonus. Any way before you replace a lot of parts and do needless work check for a hot spot in the fuel input line.
If both of these suggestions can be ruled out then the next step is to determine if the carb is percolating or the fuel pump. To do this you will have to take it for a ride and duplicate the frame work where it stops running correctly. At that time a temp check should be done at both the carb, the fuel line to the carb or the pump itself and this answer is similar to a previous post, determine the hotter spot either using a laser style temp gun or carefully with you hand. If you find the source of the problem you are going to have to heat shield it some how.