So sometimes when I use my left turn signal, my passenger side headlight goes out and won't come on unless I turn the lights off and back on again.

I can get around this issue most of the time by grabbing the multifunction stick (or whatever it's called) from the base so I don't interfere with the headlight knob at the end, however I think I did this the other day and it still turned the light off.

Should I get a new multifunction stick?

Pretty straightforward issue it seems.
Replace the signal stalk.
It's a piece of cake. 10 minute job. Been there done that.
If you can, go to a junkyard and pull one off a wreck. Much cheaper that way.

Swapped it out with a replacement.
The replacement works but I noticed my headlight was off when I pulled up to the gas station just now.

Returned home, parked in the driveway and turned my lights off then on again to get them both working. Next I put my signal on and off, left and right, and even turned the wheel to get it to go off automatically but no problems.