4 Runner Problems

I am a driver of a 1987 4 Runner. As it just hit it's 16 year mark, it had decidied not to start. When it does start, it runs around with both the Break and Battery Lights on in the dash. It has done it before, but a quick cleaning of the battery connections had solved it. This time it didn't. So i bought new adapters to the battery, and it has not worked. I just changed the starter in December, so i took it off and had that checked and found it was a bad one, so i exchanged and put on a new one. Well, it started once, and that was all. SO know i am scratching my head on what's next. Is there any ideas?

Thanks

My starter is in replacement mode. Twice now. Fails to start whenever the ambient temperature is above 80 or so degrees. NEVER fails in cold country nor any other time. We're hoping the new one (Toyota rebuild) is just defective in the same way. Frustrating. Any one else have this problem?