Is it possible for a deep sinus infection to cause a swollen jaw and the back gums and teeth on one side to hurt really bad?

Hmmm?: Pain to teeth and gums? Yes. Jaw swelling? Perhaps, but not usually. Have your pcp or ENT specialist treat your sinus infection and your dentist examine you to rule out dental problems. You many have either one or both a sinus infection and a dental problem at the same time. There’s no way to know over the internet and it’s best not to make assumptions. Better to be safe than sorry.

Likely the tooth: If i had to bet, the jaw is swollen on one side because of a dental infection, and that has secondarily infected the sinus. It could be that a sinusitis causes dental pain, but swelling of the gums would be very unusual. I'd see a dentist first for a check (asap), and if it still isn't better you may need to see an ent.

Could Be: Actual swelling (that is look in the mirror and your face is puffy) from a sinusitis is not very common, but can occur in severe cases. Dental pain due to sinusitis is quite common and is usually focused in the upper back teeth. A bad tooth can mimic a sinusitis as well...or even cause sinusitis on rare occasions. Your dentist should be able to determine the source and treat/refer as needed.