Turn signal fuse keeps blowing, hazards only work on one side

I had to remove my headlights to replace the radiator. The day before that I replaced the spark plugs and valve cover gasket. I don't think that could cause any related issues, but there were some electrical connectors and wires that had to be moved out of the way.
Anyway, upon reinstalling the headlights after doing those things, the blinkers on both sides and the hazards are behaving very erratically. The hazards don't work at all on the passenger side and sometimes not on the driver side and they seem to change speed. I just tested it again and they sped up a lot and it seemed like the dash lights were also acting up. The headlights work on both sides. I checked the turn signal fuse and it was blown so I replaced it. It immediately blew again upon testing.
Strangely, with the hazards on the rear passenger side blinker blinker very faintly, but it doesn't do so at all with just the turn signal on.
I don't have a ton of experience with this stuff, so I am hoping this info can help diagnose the issue or get closer to doing so. Thanks!


Update: Ok, so now I am somewhat concerned. I was doing more testing, flipping through brights, etc. Then with the hazards on the hazards, dash lights, clock all started blinking faster and faster until everything stopped. Then I turned my brights on to see if those still work. They don't when I turn the switch for them to stay on, but they do when I pull the stick back. And now the car won't even start. It just clicks when I turn the key in the ignition and the gauges on the dash vibrate and make noise. Before I started testing the blinkers and encountering issues, I had idled the car for a while and driven a very small amount around the driveway, so the car was operable.

More info, the problem seemed to be just the passenger side blinkers until I turned the headlights from the lowest setting to the middle one that turns on the dash lights and tail lights. Then things got weird fast.

Update 2: After charging for a few hours I was able to get everything working and the fuse didn't blow. Apparently a the low voltage from a low battery can cause unpredictability and problems that indirectly blow a fuse. I was thinking there must be a fault somewhere. I hope that the low battery was the only problem.