The first thing to consider is fresh or frozen venison?
1) If the processor is dealing with fresh venison then that restricts him/her to the life of the meat before spoiling. We limit ourselves to a 10 day maximum in the cooler at <34 degrees. We are also very picky as to the condition of the venison when we put it into the cooler, in order to guarantee that we can hold it up to ten days before we process it.
2) Frozen venison is a entirely different scenario as it could be held as long as 2 years if properly wrapped. Now you're dealing with the policies of the processor. Some processors that are doing it for a living (we do it as a hobby) take as many deer as possible in order to assure themselves of work for as long as possible. This is done not necessairly out of greed as much as it is a necessity in order for them to stay in business. While it can be frustrating to the customer, there is not a lot that can be done about it if the processor has decided to go that route.
My recommendation when taking in your venison is to ask what their "maximum" turn around time will be. If they're out more than a couple of months you may want to take it elsewhere.
As to your current situation, I would call some other processors to see if they are caught up and could take your meat. If they are open to taking it, I would pick it up from the current processor and take it to a new one.
Dave & Ruth
PS. We're caught up.