Brake lights, hazard lights, and turn signals not working

The first thing I'd do - because it's the easiest - is check bulbs (if you haven't already) and the fuse box. I doubt the fuse is blown but if the connections look dirty you could clean them with some WD40. I don't know your fuse box and it's possible some connections are always live so before you go spraying WD40 I might disconnect the battery first. If that doesn't do anything there is probably another bad connection or switch. Check around the new switch and make sure all the connections are good and clean. If they're dirty use the WD40 and tighten anything that's loose. Sometimes you can find a wiring diagram by searching Google and this can be a useful resource.

Also, this explains a bit about the wiring for a Buick Regal having similar issues and it could help: https://www.justanswer.com/buick/7q7xw-buick-regal-right-rear-brake-light-will-not-operate-turn.html