Need some help

I have a 1996 Honda Accord LX with 79000 miles on it and a 4-cylinder automatic engine. My problem is that the car keeps killing batteries. I have put two batteries on it in the past two months and it killed them both. Now, it won't start without being jumped off. And after being jumped, it will run until shut off and then won't start again. Battery cables were replaced, as were the ends. Connections were checked and found to be fine. Alternator was checked and found to be fine. Don't know if related or not, but turn signals also will not work when the air outside isn't hot. But work fine when it's a warm day. I have had two different people work on it and neither seem to know what the problem is. Any suggestions?