I have never used a .308, but understand they are excellent. I have used a .270 since 1973 and it has served me well. I have never lost a deer because of the Rifle. It's accurate and ammo is always available.
You should strongly consider buying a Tikka T3. This Finnish rifle (made by Sako) guarantees sub 1 MOA groups and costs less than $600 at Bass Pro but you can buy it lots of places. Many calibers are available so pick what you want. .270 or .308...flatter shooting...muzzle velocity...down range energy: both have their subtle advantages or disadvantages. If you can utilize the sub MOA ability of your rifle all the rest of that info is backround noise. Have fun!
Find yourself a nice plastic Tikka T3 or older m695 in .270 it is boringly accurate and is "The" smoothest action on any rifle ever built. It is the only rifle I own of which I can cycle the bolt without taking my eye from the scope/target.
Id say go with the 308 just because i like it and have used it most of my life. and if your looking for a cheap but still quality rifle get a marlin xl7 i live in texas also and took 3 deer with a synthetic 30 06. i sold the rifle to buy a 30 30, but i also already have a Remington 700 in 30 06. if you want to spend a little more get a remington 700. i also have a winchester model 88 which is a lever gun, if you can find one and its a good price get it. they are some of the best guns. the winchester model 70 is great too. i have experience with all of these guns by the way.
I don't know about hunting deer in Texas, but here in Kansas either one works very well. I have a .270 & some .308's, but in my whole life I have never owned a 30-06. That's the next deer rifle I'm gonna' buy! That & a 7mm Mag.
If you're riding' ahead of the herd, take a look back every now and then to make sure it's still there.
If recoil is an issue, go with the 308. You will not see much difference though, but some. The .270 kicks like an 06. I have all three and prefer the 06, but thats just me. About the only commercial ammo you will find for the .270 is 130 and 150 gr bullets. I prefer the 130. To get the most out of any rifle, hand loading is the key. The 308 caliber is my choice primarily because it retains energy better further down range and because bigger bullets make bigger holes.