I second your dislike of umbrella strollers. They hurt your back when you have to bend over to push them all day. Also, you can't keep much in the bags on the back of them because the stroller is so light. Every time your kid gets out, the stroller will tip over backwards because of the weight on the back. Drives me nuts!
That said - they are super easy to deal with on shuttles and trains!
I just wanted to suggest that, if you have an older kid with a stroller, you don't bother moving it every time you go from ride to ride. Rather, park it somewhere central within each land. You can ride all the rides there, then go grab the stroller and just use it to go to the next area of the park. You'll still have it for the longer walks, and for storage, but you don't have the hassle of constantly moving and re-parking.