I've had this problem since I've owned the truck. Never bothered me enough to "really" work on it till I got to this point in the "fix up things" processes.
My front turn signal lights always were weird. When I pull the switch on to the marker light position (or headlight too), my dash (turn signal) light stays on, on the right side only (pass side). When I turn on the turn signal, or hazard lights, it shows them to be working/blinking (but when I stop them, the right light still stays on, in the dash). Looking at the turn lights in front of the truck (just under the grill for '76), the right side is not working, but the side marker light is.
Here's the weird part. I changed the left side (turn signal assembly)to the right side and it works. I figure it's got to be the "turn signal housing assembly" on the right side screwed up (correct?). So I go to the wrecking yard and find two more "turn light assemblies", part #48446/3489506, bring them home, clean them up, and install them. Now I have both turn signal dash lights staying on, and no working in the front at all (turn signal, flasher, nothing!). So I put the orig. one back in the left side again. Now it's working again. Swap it over to the right side, it works there too (but the new ones from the wrecking yard will not work on either side, along with the orig. one on the right side. Also when I pull the bulb out of the right turn signal (installed in the truck), my side marker light coms on, and blinks too.
To recap:
I (NOW!) have four turn signal assemblies, correct part #'s for my '76 truck 48446/3489506. Only one works in the truck (no matter what side I put it on). The other three, I can't get to work correctly, no matter where I install them.
I've changed the turn signal "clicker" (from Rock Auto) in the steering column.
I've changed both signal stats in the fuse box, fuse too.
Continuity is perfect in all four turn signal assemblies.
I have power to the pig tails to power up the turn signal assemblies.
I show power to the bulb in the turn signal assembly when installed, but can't get the bulb to light up!"?" WTH!!!
Bulbs are new 1157A (yellow color), plus I have a crap load of older bulbs 1157's, I tried a bunch of them, nothing. Also put dielectric grease on everything.
Damn it, I'm stumped!
Here's a pic of the assembly in question.

Check the grounds as they are known for problems.then check the small contact that the bulb touches.Ive seen a lot sunk in the plastic.I usually pull them out and stretch the spring a bit to get better contact on the bulb bottom.Ive also had some that the case of the bulb wont ground well in the socket.I add a ground wire to the inside so the bulb gets better ground.Rocky

I've even soldered an auxiliary ground wire to the outside of the "socket" on the lamp. I had a 75 Ramcharger that had these very same problems and it was ALL grounds. I even ended up drilling a small hole in the lower panel that the lamps fit in and running a wire to the battery negative. That in itself solved the issiue permanently. Go figure.....

Two things you must have for a bulb to work is power and ground. Your problem lies therein, and since one assembly works on both sides it sounds like you have a socket problem.

I thank you all SOOOOOOOOOOOOOOO much for your incite, and advice. After reading through your suggestions I took the light assembly and grounded it to the - on the battery. The plug end I touched the + side of the battery. With the bulb in the socket, nothing! I then put a jumper wire on the socket and touched the grounding area of the assembly. poof, on came the light. I guess the socket part wasn't crimped good enough to the assembly that grounded to the body via the mounting screws. I took both assemblies out and put them in a vise, used a drift punch to make the crimp of the socket better. All is right with the world again. Turn signals work perfectly, lights are out in the dash when their supposed to be, and on with the turn signals.
I just couldn't believe three of the assemblies were bad in this area (even after bead blasting them). I guess we all should not "assume" anything on these trucks (cause we all know what assuming can lead to!), or cars for that matter. Thank you all again, I'm stoked now. Now on to other problems on this thing.

Im glad it sorted out for you.Most all my Rams I had to run grounds to.The rears were the worst.The housings ground thru the screws that hold them top the body.Here they rust away and the lights do weird things.Rocky