I'm in the same situation right now. Well, sort of. I'm looking for a school to go to for web design and development as well. I actually happen to have a job in web design/dev. and currently work full time managing it (I lucked out and got the job because of my experience and no degree), so I'm trying to figure out the best option for school while I work. I'll probably have to do some kind of online program, but I don't want to do the Full Sail thing either (I heard that people who go to Full Sail struggle to get jobs).
There are some schools that offer programs specifically in web design and development. I am looking into the International Academy of Design and Technology(iadt.edu), which seems to be an accredited US school. I'm not completely sure about it and have no idea what their program is like, but they do offer an online bachelor's degree in web design and development.
Of the designers/developers that I have met, it seems that most of them have at least a bachelor's degree, but it may not be necessary if you have the experience and the right opportunity comes your way. It sure helps, though.
What I can say is that you're definitely going in the right direction wanting to be able to do it all. I can't tell you how frustrating it is to have designers that only know how to do one thing. In the past, we've had one designer make the psd, another slice and code, and a separate developer code javascript because the other designers don't know how. It is a huge pain to have to send files back and forth between everyone and hope you have the most recent version when you show it to a client. You'll do really well if you can do everything.
Hopefully this helps.