I am not 100% sure on how these things work, but I was hoping that since my car trade in value is 7500 and the car that I want is 6700 I could just do a straight swap and leave my loan intact and just continue to pay it off for the next three years
That's not going to happen so put that out of your mind. It seems like the most cost effective method for all involved (you, the bank, the car dealer), but 2 of those entities aren't logical beings--they are bureaucratic organizations with rules and procedures and paperwork. Balances have to be zeroed out, liens have to be released and issued, titles, bill of sales, etc, etc.
Usually what people (and by "people" I mean "people with poor money management skills") do is roll over the amount they are upside down to a new car loan to purchase the new car. That's option 1. Option 2 is to get an unsecured loan for enough money to pay off the underwater amount and buy a new car--this will probably be a higher rate, but you get a lien free car and its a little less hassle at the dealershop.
However, with all that said--the best bet is to post a case study and get your finances in order, because this shouldn't be a crisis with your income--even the low-end income (56k) you have.