Well, the CPAP machine will just tend to help keep positive pressure of the air that you breathe in during the night, to help keep the airway open (particularly at the posterior pharynx, where you are probably occluding the airway, causing severe snoring and contributing to the sleep apnea).
There really should not be any relation to CPAP to worsening of sinus infections as this CPAP should not be introducing bacteria to the sinuses, nor impeding them from continuing to naturally drain. However, people with sinus infections can feel congested, and the CPAP may feel more uncomfortable for them. Of course, some people may swear that using the CPAP worsens their sinus infections, but it probably is a coincidence -- that these certain people have problems with their sinuses in the first place -- as many thousands of people have used CPAP without issues.
Should I do the titration study if I have a slight ear infection right now? I went to my regular doctor a few days ago and he gave me amoxicillin for it but it doesn't seem to be working.
It would probably be OK.
Of course, if you are having a lot of ear pain, it probably would be best to wait, since theoretically blowing air into the nasal passages via CPAP during the sleep study could theoretically increase ear pain (via the eustachian tube, which connects the middle ear to the back of the throat, which is how you can "pop" your ears while yawning).
And since the sleep study is certainly not an urgent test, if you're feeling poorly at all, you might as well postpone the test.