I've heard good things about PB Blaster, and I use the Liquid Wrench I have on hand for stuck parts. Patience and a little heat will help too. I found both PB Blaster and the bottles of Liquid Wrench at Lowe's (try Seasonal or the Tools sections), so you could probably find those at most any hardware store. Is there wiring inside that is messed up? Be careful because some of the cleaners and lubricants can dissolve rubber and plastics.
On the tension adjustment for the brown Kenmore, it looks like it is the lever on the front of the machine just directly above the needle bar. Be careful if you remove this part to clean it. The flat tension disk mechanism on the side is very likely held in place by the tension adjustment. The first time I took off my adjustment dial on my White to polish up, the whole tensioner fell off.
It wasn't difficult to fix once I figured how it was connected, but it was a surprise!
If you think you'll have to pull it all apart to clean well and de-gunk, see if there aren't a couple of screws towards the inside of the "head" part (the part with the needle bar, presser bar, and tension). These screws would probably be facing the pillar of the machine. If so, these just might detach the whole "head" of the machine, needle bar, presser bar, and tensioner all. Then, you could clean and oil as 1 large piece if you didn't want to take all the individual pieces off. Of course, doing this will also make it easier if you want to take off the tension mechanism and lever part to clean them thoroughly. Kind of depends on how handy you feel! I'll look around and see if I can find a picture of what I'm talking about.