Maxillary Molar?

[Mony]

Ok about 6-7 years ago, I broke my maxillary molar(first top molar) in half exposing the insides. It hurt like hell, but I took the pain and it gradually went away. After a while of having this tooth fully open and exposed it turned black along with the gum in and around it(sign of infection occuring). Well my breath started getting real bad and i started having a yellow coating on my tongue, although it's not as thick as some i've seen on the internet. Well i read that the maxillary molar is really,really close to the maxillary sinus, and that it is possible for a M.Molar infection to spread to the M.sinus, thus causing swelling in the sinus. This swelling then causes congestion, which i have for most of the day, and causes excess mucus. The excess mucus then goes out the back of my sinues (post nasal drip) and collects in the back of my throat. This lets the anaerobic bacteria to have a food source to feed on all day and lets them produce their sulfuric smells and cause my bad breath and yellow tongue.

My theory is, if I have this molar taken out,let my gums heal and take antibiotics to cure any infection in my nose, that it will inturn reduce swelling ,relieve congestion, and stop excess mucus production, not allowing for the anaerobic bacteria to feed on the proteins in my mucus all day. This will, i hope, cure my bad breath.......what do you guys think.