OP,
I've been down a similar road recently and perhaps I can help. My wife gave her Mazda Protege to our son and wanted a new car to replace it. We looked at 5 models: the Civic, Corolla, Sentra, Elantra and Sonata. After visiting each dealership we had a clear winner (for her): the Civic. Dollar-for-dollar it is the BEST car in the group. She gets about 35 MPG in city driving and loves it. Unfortunately for me it was the most expensive in the group but she's happy so I'm happy.
It seems, from your post that you are torn between the Corolla and Civic. My opinion (if it even matters) is that they are two very different cars. My local Toyota dealer had a "take-it-or-leave-it" attitude ... and the price that you mentioned ($12,500 with dest) may be a mirage. They were selling for in the $17,000 range with no negotiation just last year. I doubt, even with the current state of the economy that you can get that good of a deal. It is WAY out of the ballpark of the numbers that I remember.
How do you get the best price? Easy ... just go to the man website for each car that you are interested in, locate all dealers within let's say 100 miles of your house and email them with what you want.
Ask for an email quote in writing.
Sit back and relax.
Take the lowest price or work the dealers against each other. In my case a dealer 75 miles away had the best price ... when my local dealer asked if I had gotten a better quote I told him and he matched it. Problem solved.
BTW I paid about $15,300 for a 2007 Civic LX ... $17,000 OTD. It was by far the best "bang-for-the-buck in mid-2007. You might be able to do better these days.
Good luck and let us know how it goes ...