Question
I'm trying to figure out what kind of grass to use and if it should be sod or seed.

I just bought a foreclosure in Northern Texas back in October and one of the biggest problems with this house is the lawn. This is my first house and I really don't know anything about lawn grass. I've been told that this was one of the prettiest lawns in the entire neighborhood a couple years ago, but the lawn is so bad now... It's an even mix of grass, dirt, and weeds. I believe it originally had St Augustine.

My biggest problem is the back yard, which slopes away from the house and has a fair amount shade. Most of the grass is dead, except for the shady spots. However since there is no grass in a large portion of the back lawn, rain has washed away much of the topsoil due to the gradual slope. I have decided that the back lawn needs to be sodded to help build the dirt back up since there is almost no grass, and I would prefer to put seed down in the front to keep the cost of this project to a minimum.

I'd like to have grass that is thick/fluffy and can withstand walking traffic relatively well.

My thoughts are to sod the back yard with Bermuda, over top of the St Augustine that is barely hanging on. Then I could just put Bermuda seed down in the front yard since most of the topsoil is still there. Does that sound like the best thing to do?

I'm concerned that the Bermuda sod will not properly suffocate the St Augustine in the back yard, and that the Bermuda seed will not be able to overtake the weeds and St Augustine in the front yard.

Your opinion would be very helpful!

Answer
To answer your questions helpfully I should have some experience of the weather conditions and horticultural practices in the area where you live. Alas, I have to confess to ignorance here.
My best advice is to ask the Extension Service in your county.
As far as I can tell, you have to first address an erosion problem in the backyard. Sodding with an appropriate grass or planting a quick growing ground cover are two possibilities.
A turf that is 'thick/fluffy and can withstand walking traffic relatively well' definitely requires the knowledge and experience of someone 'closer to home' than I am.
I am sorry that I cannot be more specific than that and finaly, please accept my apology for being so tardy in replying.