My car requires catnaps

Hi, guys: My '92 Buick LeSabre has been to the mechanic twice without resolving this problem completely. I have a new battery in it. When I go to the store and am in there just briefly, the car will not start again. It may turn over, but if I keep trying the battery will go completely dead. If I call AAA, in the half-hour it takes for a tow truck to come, somehow the battery has regenerated itself, and the car will start. So I’ve learned not to bother with AAA but just to factor in the 20-30 minute catnaps along the way. However, after I’ve made several stops including two or three catnaps, then the fan stops working. I’m afraid that the power steering will go out while I’m driving; that was the problem I started out with. What can be wrong that my mechanic isn’t finding? I don’t drive much any more; is that part of the problem?