Should I get my transmission replaced at the dealership?

I have a 2007 Honda Fit Automatic (Sport) with about 80k miles. Transmission has gone bad, so we took it in to the dealership to have them look at it. They checked the transmission fluid which had turned black by this point & said that something was definitely broken.

The options they gave were $2800 for a used transmission w/ 1 yr warranty or $5500 for a new transmission w/ 3 yr warranty.

I haven't gotten a quote from any other auto shops in the area - no clue which ones are good, but I realize I have a chance of getting the work done at least a little cheaper by not using the dealership.

We also have the option of trading in for a new or used car (probably another fit). If I go with that option, should I suck it up and get a manual rather than automatic? Does that make much of a difference in terms of how long the transmission will last?

So should I just have the work done on this car? If so dealership or other garage? Or would it be a better use of money to get a different car?