It sounds like you’ve got two separate, but related problems. Logically, both signals can only flash at the same time if you energize both circuits at the same time, in effect turning the signals into four-way flashers. The only way that can happen is if the energized signal shorts to the non-energized signal, which can happen in the wiring harness, or in the switch. Apparently, you also have an occasional short to ground in the same circuit, which is what blows the fuse. Since the most likely suspect is the switch, and it’s where all the hot legs and grounds are in closest proximity, that’s where I’d start my search.