Your mistake was going to your bank, it is very hard to qualify for loans or refis with the larger banks right now. You want to get qualified with a Mortgage Banker, they are Direct Lenders.
You need someone to look at everything you are wanting to do and find the right direction for you to go in. Refining your home will depend on the type of loan you have and the current market value.
Your payment seems very high. Your property taxes may be higher than they should be. If you want to talk about your situation, please give me a call, so I can get a better idea of your needs, then I send you to the right person, Call me at 916-769-3694.