No turn signals when lights are on

It's not the headlamp switch or the flasher. Most likely it's one or more problems with faulty grounds, faulty sockets, faulty wiring, and/or faulty bulbs. The easiest thing to try first is to put in four new exterior bulbs. Use regular 1157 bulbs in back, 1157A or 1157NA in front. Pick a reliable brand (GE, Philips, Sylvania). What you're hoping for here is that one of your old bulbs has dim and bright filaments shorted to each other, or trouble with their base shells or contacts—or that somehow a wrong bulb (a single-filament item such as an 1156 or 1141) somehow got crammed into one of the four sockets, and its single base pin is bridging the two contacts in the socket and creating the path for the feedback you're describing.