Wait defore taking the column apart; I'd bet you have a bad bulb on 1 side (maybe more and a bad flasher. A tip on flashers; Many factory ones are "color-coded"; a green one is designed to flash "X" number of bulbs at such a rate (frequency or second or maybe per minute), a blue one, a yellow one a pink one, thye are all different. Each bulb represents so much resistance in the system and too much resistance makes them act goofy. Anyone who has ever towed a trailer and wondered why their turn sigs work goofy has seen this.
The aftermarket ones are "heavy duty"
Also the not blinking is a "safety feature" they built in so you know to check for bad bulbs. from the factory since they started doing the color coding they ALWAYS use a different flasher for the 4-ways from the turn signal. (twice the number of bulbs) the 4 way one will work fine in the t/sig slot but not the other way around. Any one you buy at Wal Mart, or the parts store (most common number is a 552) will work in there. There are 2 prongs on them at a 90* angle. They always said to replace these with a heavy duty version for towing anyway.
and most of the replacement ones, won't "not" blink with a bulb outage.
I'd bet a bulb (or a few) and a flasher, maybe clean up a bulb socket here or there and you will be good to go. Remember the rear tail lights are a dual filament setup. ("2 bulbs in one") Park lite on one and brake/turn on the other filament. so your park lite may work, but the other 2 not work.
On my truck I gotta cut/solder in replacement pigtails on the driver rear, especially on the b/u lite since the socket is so corroded it won't hold a bulb. License plate lites too, cheap excuse for the cops to go on a "fishing trip"; they are great for using that as the reason to pull you over, possibly to wind up costing you much more with the "Oh by the way's" and write you up for other thing(s). Ba$tards.