The bi-metal tab in the flasher heats up faster anytime the current increases. This causes the flaser speed to increase when hooking up a trailer. I am with dennis I'm not sure how a missing or burn't out bulb could cause the thing to speed up, but anything is possible.

As far as the LED conversion is concerned, the flasher will usually stop completely because the current is so low that the bi-metal tab cannot heat up. There is a solid state flasher that is available to cure this problem. They have been used on RV's for years and can be found in most RV supply houses.