2005 Neon SXT 125,000 mi.
My turn signals and dash indicators worked during a drive to a highway then didn't after driving 32 miles. The hazard lamps and dash TS indicators flashed when actuated and the rest of lamps/wipers on the stalk function normally. Drove home using hand signals where drivers passing by waved back at me. Got on my desktop to pull up the factory shop manual and see what others experienced here and other forums. Didn't care for the idea of buying parts nor tracing wires so went back out to the car to horse around with the stalk. Got it running then turned on the hazard lamp. Pushed the stalk up for a right TS then turned the wheel to the left to release the TS switch. Pressed the stalk down and the TS flasher started working again. Turned off the hazard lamps to find both signals working again, including the dash indicator lights.

I made 7 or 8 turns when driving home and used the stalk each time out of habit before using hand signals. The only thing different this time was having the hazard lamps flashing when I tried using the TS. Is this an indication of a failing flasher?

 

Maybe the flasher, or its contacts acting out. Next time first try thumping & or wiggling the flasher to see if it'll wake up. If so, unplug it & check its socket & pins for corrosion, loose fit, bent pins, or spread sockets.

If dirty, try CRC plastic safe MAF spray cleaner to tidy things up.
If sockets & pins are ok, open that puppy up & check the inside solenoid works for rust or corrosion & its electrically switched contact points for wear, or debris & tidy them up with the plastic safe cleaner, or try cutting & pulling / sawing back & forth, a border sliver of a New / clean dollar bill, between the contacts to clean & freshen up Both contacts surfaces. Don't touch the sliver of the New dollar bill with your fingers, so not to contaminate the contacts with body oils. Maybe dampen the sliver with just a whisp of the spray cleaner as you pull it through the closed contacts, so you clean Both contact surfaces & they dry, clean.

I've also used 600 -1000 grit Wet type sandpaper to freshen up & Wake up my 94 Taurus drivers side master control power window & door lock switches with this technique & its been a long lasting success.

Interesting, humorous happening:
A TCCA member reading my contact cleaning post & not having fine grit wet sandpaper, nor a New dollar bill, used / cut up a New Ben Franklin / $100 bill to clean his!!!!
BUT he was Careful not to damage the bill Serial Numbers when he cut the slivers, so when finished he just took the cut up remains that had the unmolested serial numbers intact, to his bank for replacement. Now thats a testament of Faith in reading something internet posted & trying it out!!! lol

Yes it worked & he didn't have to buy the expensive replacement switch assy the Dealer was trying to sell him & Yes the bank replaced the cut-up $100 bill. Don't think I'd have cut up a benny, but I'd have likely tried the fine grit wet sandpaper & plastic safe spray cleaner & non linty foam swab to tidy up the contacts.

 

I'm an old hand at cleaning contacts since it's cheap and easy plus often corrects the problem.

I reread my original post and noticed had I left out details. There wasn't a click when the stalk returned to center when turning the wheel cancelled the signal after the lamps stopped working. Then I pushed up on the stalk with the hazard flasher engaged with it staying in emergency flashing mode then heard the click when turning the steering wheel to cancel the signal. That's when I turned off the emergency flasher and tried the TS again to find those now worked normally again and clicking when released.

I guess some gizmo hung up inside the housing that sits on top of the column. Can that be opened up or is it a sealed unit?.

 

I don't rightly know what it looks like in there as I've never had to open ours up, thank goodness!!! Maybe someone reading this that has will chime in & enlighten us on what it looks like in there.
You might take a look in the shop manual link atop this forums thread index page here 2nd Gen Neon Parts Manuals - DodgeForum.com & click on #2 post of moderator darthroush Red color FSM year range link that covers your year / model for the stalk assy exploded view to get familiar with the assembly parts & how they go together. Maybe all it needs is a spritz or two of CRC Heavy Duty Silicone spray to unstick & loosen things up some??? More thoughts for consideration, let us know how it goes.