Now that my cat is getting allergy shots will sinus infection clear up without operation?

I was told recently that my cat's cat scan showed an advanced sinus infection and she needs an operation. We took her to a specialist, they are all internist and oncology animals vets. They may be right, she may need this but I don't want to do this to her. She's 8. She just got over Pasturella and is now receiving allergy injections as of today. 1st question: Are the allergies related to the sinus issues? 2nd question: If we wait until she has about a good month's worth of injections is there a chance the infection will clear up? I just don't want to rush into this and its horribly! expensive, would wipe out my savings! HELP PLEASE. The poor cat is miserable. She's been on antibiotics already but I know they are dangerous if used for extended periods in fact it could kill them or give them sever problems in organs.