I have a 6 millimeter overbite and my orthodontist recommended temporary Anchorage (TAD) device but I do not want to do that?

When the braces are on and they are trying to pull the front teeth back to correct your overbite, they are actually pulling the back teeth forward as well.  (Like a tug of war)   All teeth move when the braces push or pull on them.  If the orthodontist wants some teeth to move more than others, he has to have some way to hold the teeth that he does not want to move.

I would guess that you probably have about 4-7 mm of space if your premolars were removed (Premolars are about 7 mm wide, but you may have lost some of that space if they were removed a while ago and teeth drifted)   So,  If you want 6 mm of movement  in the front teeth, you have to hold the back teeth in place so there is  almost no movement of the back teeth. If you have less than 6 mm of space,  you have to close the space and then move ALL of the teeth back to eliminate any additional overbite.   TAD's are the best choice if you are not willing to wear a headgear (in my experience, no adult will wear a headgear enough in a case like this) and the ONLY choice if you have less than 6 mm of space.  The fact that your orthodontist is thinking this way indicates that he or she is probably good and you will get a good result.  

Why does this make me think your orthodontist is good?   Because Orthodontist's know that  nobody really wants TAD's,  and honestly, orthodontists would rather avoid the hassle of putting them in.  Also, Talking about TAD's sometimes freaks patients out or causes them to go to another orthodontist.   But TAD's are the state of the art standard of care.  Your orthodontist is actually concerned about getting you the best result and the only way to do that is with TAD's.  He or she has gotten the extra training, purchased the extra tools and supplies, trained the staff and set aside some special days each month to put in TAD's.  Not because it is profitable or because it is fun to do (yes, it involves in injection and most of us got into orthodontics to get away from that part of dentistry) He or she is doing it because it is the best treatment available and they want to do what is best.

He or she is putting your best interest (completely fixing a large overbite) above their best interest (increasing his income and starting the case by telling you what you want to hear).  This is the definition of a true dental professional.   You could probably find an orthodontist who would happily treat you with out TAD's, charge you almost the same fee and only close half or less of your overbite.  You could then join the others on this web site with posts like "Braces almost done and still have over bite" or "My teeth still stick out after braces",  of "Teeth slant to far forward even after braces" or something like that.  

Stick with this orthodontist and pay whatever price he or she has quoted you.  You are probably getting more than your moneys worth.