1987 Buick LeSabre intermittently stalls

Here is another issue that is bothering (The Beast) my Buick. Okay, Lately,The car starts up fine-no real issue with starting besides the mild shaking I feel. I know that back up north in NJ I had an issue with the idle going up and down at start up, but after a couple of minutes she would smooth out. Now, in Florida I start the car; she starts up fine, but after 5-10 miles of driving and then come to a complete stop (engine off)like when I have do some shopping in the local Walmart. When I return to the car and start (The Beast) up, the car starts a light stuttering, like to much air getting in and then stalls, but if I let it sit for about thirty seconds she starts up fine and I’m on my way. I got a couple ideas,but would like hear what my “STAY DIRTY” comrades have to say. :stick: