Thanks for all the suggestions everybody! Although I really liked the idea of going to a car detailer or upholstery/dry cleaner, various factors made that impractical. I decided the next best solution would be to do as mmichel suggested. All the screws came off no problem and I tossed the fabric in the washing machine. It took three washings and stain removal applications to get the vast majority of the gunk off. I always made sure to use a low spin cycle so as to not stress the material too much. After air drying, I put the fabric back on the case. This is where I started to get a bit nervous because the fabric had partially lost its form and was difficult to get back on the wooden frame. I decided to push on despite nearly all of the screw holes in the fabric not lining up with preexisting holes in the wooden frame. There were some spots (for example, where the fabric is shaped around the saxophone bell part of the frame) where the fabric was very taut after putting in the screws.
Now, more than a week later, the fabric is still taut, but I haven't noticed any tears in the fabric or other signs that the fabric is too stressed. I feel quite confident that everything is going to hold and be just fine. One thing I'd be conscious of is taking off and reapplying the case fabric multiple times as this might cause the wood to weaken due to having several different screw holes in close proximity, but that is just a guess on my part. If anything changes as time passes, I'll be sure to post an update here. Cheers!