1995 YJ turn signal works but then stops with non-blinking green light

Like Chief Hof indicates replace that flasher unit. Id also pull out all the blinker/tail light bulbs and throughly clean the sockets and bulb ends. Corrosion and years of dust make for some bad connections and things can get problematic.
 
No worries...as those flasher relays wear out they can overheat and cause internal shorting issues...make sure you get the correct relay, they come in two or three prong configurations and are around $10 a pop...let us know how you make out...
 
Mine does that left and right. If I tap the signal lever after it "cancels" (that means goes to solid non-flashing turn signal after it tries to cancel) it turns off as advertised. I noticed that the grease on the mechanism was old and almost gone last time I had it open. I will need to pull the steering wheel again and open it up to lubricate everything in there... I think a thin white lithium grease is needed. I think that will fix it.
 
How do I remove the flasher for the turn signals and hazard? They won't just pull out and it doesn't seem like there is much behind them to plug in that way . I have a 95 yj jeep 4.0 wrangler
 
How do I remove the flasher for the turn signals and hazard? They won't just pull out and it doesn't seem like there is much behind them to plug in that way . I have a 95 yj jeep 4.0 wrangler

They do simply pull out, toward you. They have been in place for most likely, 24 years so they might be comfortable where they are lol. Just pull on them, they will come out.