I can understand that the condition of tour gums is very bad. Usually, gum infection leads to infection of bone and eventually bone loss occurs. When this happens, teeth loosen due to reduced bone support. Normally, these loosened teeth are held in position by the accumulated deposits (calculus or tar tar). When we remove these deposits, teeth shake a little or they become mobile. Most people think that the teeth have become weak because of the cleaning procedure. The fact is that the teeth are already loose. But cleaning has to be done because the deposits will keep destroying the bone and teeth will loosen further.
Next Steps
See a Periodontist immediately. A full mouth x ray might be taken which will show the extent of bone loss. You might need deep cleaning (cleaning from inside the gums) under local anesthesia, or if much worse, you will need a flap surgery or gum surgery. Either way, get treated at the earliest because, if you delay, you risk losing all your teeth in a few years.
Health Tips
Mouth washes, tooth pastes, gum massage gels...all of these will not work as long as there are deposits inside your gums. These things will have additional benefit only when you get the deposits removed. It is like not removing the source of infection from the body and taking medicine for fever. As soon as the effect of medicine wears off, fever will start again.