If the door handles have been replaced, that may be at least part of the issue, as smontague was saying. The leverage on the inner paddle is less than the originals. Probably patterned after a Mustang or Falcon or something like that.
The fact that yours are just getting worse as you try them means that they are likely very gummed up as the others were saying. But you need to be careful with carb cleaners as they will leak out the seams and get on the paint. On old paint you have some time to clean it up, but on fresh paint you could be toast in seconds!
And I have found that when they are this bad you may not even be able to fix them with standard spray cleaners. That old lube that was used is just too old and gummy and resists even the most aggressive quick-solvents sometimes. A lower acting de-greaser might work better, but a good soaking in a tank is the best.
If a quick shot of whatever doesn't do the trick then, go to the extra trouble of pulling the mechanism out (one door at a time only though) and giving it a good thrashing with some super degreaser like that Super Purple stuff, or maybe an industrial strength 409 (I've never had good luck with that myself, but some swear by it) or some form of Chemtool or whatever.
If you have to manually scrape some of the goop out, go for it.
The mechanism's look pretty complicated (like a clockwork at first) but it's actually pretty basic. Easy to see if a spring is buggered, or a bar/tab is out of line, or gear bent. Easy too to fix up to a point.
At some point you just need to replace the whole unit with a new one. But the originals are very robust and can still work exceedingly well when cleaned and re-lubed.
If you clean yours up and it looks good, but some boat trailer, or other water resistant grease (I use synthetic) or even just a light spray lube that leaves a film. I prefer grease, but it does tend to gum up over time (years) and some might prefer a lighter lube that does not collect junk. I like grease though, and it's worked for me so far.
If your mechanicals are still original and in good nick (as Edd would say) then I'd clean and reuse them.
Sounds like a lot of work, and it can be. But it's totally worth it. And leaving one door intact while working on the other is like drum brakes. It's great to have a reference point when you're trying to figure out where the rods go!;D
Paul