The problem with shaving such an old cat is that if the vet does it, they would usually put her under while they do it. This is a big risk for such an old cat. If you have the equiptment to do so and she allows you to do it, then go ahead and try to shave her. It would be best to do so because the mats are painful and uncomfortable. Unless they clipped all the mats out and she has no more but JUST looks bad. Then I wouldn't worry about, dont put her through that stress if it isn't really necessary. After awhile she will look just fine and her fur will be even again.
Also, that is really awesome that you adopted such an old cat. Most people don't want to adopt cats over 5 let alone 15. I adopted a 7 year old himalayan from the Humane Society that I worked at and he was also badly matted to the point where the mats were just one smooth layer covering his entire back. He was shaved but they didn't do a very good job and I just let him go and after a month he looked a lot better and I clipped his fur a bit to even it out. No I just occationally shave the under side of his belly and he is ok with me doing that. He doesn't groom well or let me brush much under his arms and his belly, so shaving is sometimes necessary.
Good luck with her.