You may dislike dealing with banks, but it's worth a single phone call to the bank that holds the current mortgage. Assuming you've been a good customer (no late payments), they should be more willing to deal with you.
Explain that with mortgage rates so low, you'd like to refinance the loan—otherwise, you're considering simply paying it off entirely.
I had a similar situation, except the value of my house had dropped so much that I was underwater. (I was shocked by this, as my house is quite modest. But our area was hit hard by the sleazy mortgage merchants, and all the foreclosures have cut property values in half.) Most of the banks I called would not even consider talking to me. It was frustrating.
When I was able to talk to someone at my mortgage bank, a different conversation happened once I said that if I wasn't able to refinance, I would probably pay off the loan entirely. I don't know if it was the threat or the fact that I had ready money, but all of a sudden they were more creative.
In a month I had refinanced at the lowest rate around. I took a shorter mortgage than the original (I had 17 years left on the original mortgage) and closing costs were minimal. The biggest cost was the appraisal.
For those of us who don't fit the typical financial profile (self-employed, financially independent, etc.) it's important to have a good relationship with a bank. Made all the difference in this case. (Of course, I got lucky when I originally applied for the loan, as we were in the wild years of "no dox" loans. Crazy times, but it made my life a lot easier!)