If they're just dim, that's one thing - a relay might improve things. With what you're describing, it's possible there's some sort of small short that's causing the overheat of the switch, if in fact that's what's happening. If so, a relay probably won't fix the problem.
You might try disconnecting the headlights, pulling the headlight fuse, and touching a test light between the two fuse terminals to see if it lights. If so, you've got a short to ground somewhere. If not, you might still have a short, just not a big enough one to light the test light. If that's the case, try touching a volt-ohm-meter across the fuse terminals in amperage/current mode at the largest current setting possible to see if there's a current draw.
If so, there's a possible short somewhere you might have to track down.
That's as far as my thinking goes...