This is unfortunate, and I admire your tenacity. I'll start by first asking if there are any other gyms in your area? I recognize that the location of Gold's is good, and I imagine that finances are an issue, but sacrifices may need to be made if training is important to you.
To your question - you won't die or quit getting stronger if you don't clean. The set up of the program will be less than ideal, however many folks have gotten plenty strong with minimal cleaning. With the plate size discrepancy, have you considered working from the hang and slowly move the bar down as you become accustomed to your new equipment?