My passenger side blinker light comes on when I put my headlights on. It doesn't flash, just stays as a steady light. It does blink when I move the turn signal lever though. Could this be something with the bulb, or is it something else?

Sounds like a bad bulb.

It's one of two problems.
One is a bad ground connection on the right-side cluster plug-in.
But it's more likely that one of the lines on the circuit board on the back of the cluster has a break in it, causing the cluster illumintaion lights to 'ground out' through the pass turn siganl indicator. You'll have to remove the cluster and check all the lines, especially those around the dash lights and find a way to jump the break.
Here's how I repaired the cluster from my 84Z. The circled areas show where the breaks were...