Go to WalMart (even Safeway sells them here) and get yourself a lingerie bag. It is a fishnet type bag that zips open and closed. Put one or two of your shirts in it and wash them in that. This keeps the straps from getting tangled in anything, guaranteed. They can be dried in the bag as well. (or line-dry them). Some bags are small and some are larger. The average will probably hold two.
My mom says that they used pillow cases in her day. Hey, why not? These bags are also good for delicates and other items that have velcro or ribbons/straps that get tangled up or caught in other things and ripped. I use mine for bras, ace bandages, bibs with tie-straps or velcro, etc.
As you say, the only problem is that you can probably only fit two at a time. You can find fairly large ones. I have two and the largest is about 22" X 20". I can fit two sun-tops with spaghetti straps at a time.
Hope this helps.