Car doesn't start sometimes and sounds "weak" when it does.

What is your question?

I bought a new battery since it was old but that didn't solve the problem. I took the care to a mechanic but it always started for them so they couldn't help. That was this winter. It's gotten worse. Initially it just didn't sound right when starting the car, weak, like the battery may be draining...but it's now got a new battery. A couple of years ago I had a new starter and alternator installed so that shouldn't be the problem but it could be. I'm wondering about the ignition switch. The lights on the dash aren't dimming nor is the dome light. There is no "clicking" as with a starter issue. The battery doesn't seem to be not charging. When I get to the 3 position (starting) on the switch it either starts quietly but immediately or it just does nothing. It's not grinding, trying to start. The dome light isn't dimming. I don't know what the problem is. I'm hoping if I take it in a mechanic can get it to malfunction. What is the most likely problem?

What makes this problem better or worse?

It's worse if it's just been driven.

How long has your 1994 Toyota Corolla had this problem?

It's been worse this last few days.