Engine cuts out while driving and often will NOT restart again until completely cool

When I drive the car for the first time each day it is generally fine but if I park it for over 30 minutes it very often will not start again. The starter and engine crank normally but it does not catch. Or if it does start, after a few minutes of driving or idling the RPMs will jump and randomly drop to zero and often the car will stall (though sometimes it manages to recover after a second or so). When this happens it sometimes restarts and sometimes will not start again for an extended period of time (up to 45 minutes). We discovered that when this happens there is NO spark to the plugs! Then after a random amount of time it will start right up as if nothing was wrong, until the NEXT time. To add insult to injury my battery is on it's way out because of the extended cranking attempts and I don't want to replace it and risk ruining a new one until I fix this issue.

I brought it to two mechanics and they are stumped. Plugs and wires were replaced. Fuel filter swapped out, intake cleaned thoroughly. And after working on the car for several hours he was SURE it was the distributor (or a component inside) or faulty distributor cap, but we replaced both (and tried both distributors with both caps) and the problem persisted. He tried to find faulty wiring or a loose ground but nothing was apparent. The car generally runs perfectly except for this issue and I am afraid to bring it to another mechanic who will just start replacing parts until he "finds" the problem costing me more than the car is worth.

Any ideas?