For whatever reason, I seem to be a stray dog magnet

Of course, all these 'free' dogs usually come with nice vet bills from getting them back in shape and healthy, and as much as I really could use the money in other areas of my life, I guess i just feel fortunate to be in the situation where I can do something to help them. In your sig it says something about Texas, but your location doesn't say Texas

If you're in Houston, I'll give you the info for the vet I use, he's very affordable compared to others.
My latest 'foundling' had mange. I came home from work the day after returning from a trip, and thought there was a dead/rotting cat under one of my parked cars. Turns out it was this little dog
You can see how bad her skin looks, but I honestly regret not taking better close ups of her condition. She had Sarcoptic mange, which is caused by mites. Good news: treatable fairly easily, bad news: highly contagious to both other animals and people. Her skin was pretty much hairless on her legs, belly, and around her ears... it was very leathery looking, especially around the ears. She had raw open sores on her legs and body from biting and scratching (it's insanely itchy and uncomfortable for them). I thought she had a broken leg, because she was hopping and not putting weight on one of her back legs (both looked bad actually... I figured she had been run over by a car) but it was because the pads of her paws were literally peeling off because of the mange... they were red and raw... looked like hamburger meat... it was gross, and obviously EXTREMELY painful.
She ended up with her little arsenal of meds, she had antibiotic shots, steroid shots, some stinky sulfur bath dip, and antihistamines to help with the itching. Most of that was to treat the secondary problems with the skin, and the actual medicine to kill the mites was called Revolution, and it was applied to her back, much like the advantix liquid stuff for fleas/ticks/mosquitos. She has been getting that every 2 weeks, but then when I thought it was better, I skipped, and noticed her itching again, so she's back on it. It apparently takes a while to clear up totally, and my vet said that's the biggest problem, that people stop the treatments too early and it comes back again.
I found her the day and a half before the hurricane hit here in Texas (Sept 10th), and here are some pictures of her about a month after me having her
If you are standing behind her, one of her spots makes a heart
Anyway, there are different types of mange, as others have posted. If other dogs are getting it through contact with yours, I'd lean towards it being what this one had. The treatments really weren't that pricey, but my vet honestly totally hooks me up with good prices because I take in rescue dogs and take care of them. Look for vet clinics rather than vet offices if you're tight on money, also check with the SPCA or call some rescue groups in your area and see if they have any advice for more affordable treatment.
What your dog has is torturous though, it is so itchy and uncomfortable for them, they will literally chew holes in themselves, and you NEED to step up and find a way to take care of it. I'd limit where they dog goes to keep it on tile, and I wouldn't touch it too much, plus really wash your hands well after you do touch her. And keep her away from other animals.
(sorry for the worlds longest reply... but I figured I just went through/am going through this, so I figured it would help)