I would be very surprised if any 3 year old cat of mine would need a dental cleaning - although it would not be an impossibility, just improbable.
If I were in your position, I would start by finding a competent Veterinarian. I would first look for a cat-only clinic nearby. This website
may help you. It is a list of Veterinarians who come recommended by "owners" of chronically ill cats. It also has some other Vet-finding resources.
Then, when you have her annual health checkup, you can ask about any need for dental cleaning. It may be that beginning and continuing regular cleaning at home would take care of current tartar buildup. More about that part below...
I would NOT advise you to base any decision on price alone. Prices will vary from clinic to clinic - but, so will the quality of the services provided. Vets are not "created equal" and, especially in this aspect of health care, you will most likely get what you pay for if you shop by price alone. (On a recent thread here at TCS, a poster identifying themselves as a vet technician, stated that several, now standard operating procedures in feline dental work, are not carried out at their clinic!)
I'd sugget that you first learn about what is prudent and advisable in any dental procedure (e.g. you already have a sense of concern re anasthesia). There is an excellent series of videos on this site
with a Veterinary specialist explaining each part of the procedure. Check out numbers 1,6, 11, 13 in particular.
You can also read that same checklist at this link
. If it turns out that a cleaning is required, I'd suggest you discuss each of those points with the Vet and ensure that each is carried out.
Now, about the do-it-at-home part....Cornell University College of Veterinary Medecine provides this set of videos
showing how to get into doing it. "Getting into" are the operative words here...it will mean that, very slowly
, you introduce the process to your cat. Be prepared for the use of lots of patience as you de-sensitize Sweetie to any concerns and teach her that this will be a pleasant experience. If you're successful with this, she'll come running when she sees the toothbrush!