Can I vote without a Texas license/ID?

I hope this isn't a stupid question. I'm an out-of-state college student in Houston and would like to vote as a Texas resident. I just sent in an a voter registration application. However, I heard that actually voting requires some form of Texas ID. I have an out-of-state drivers license. For personal reasons, I'd rather not exchange my current license for a Texas drivers license. Will I be able to vote without getting the license?

Here's another potential issue. I was registered to vote in a different state, but I never actually voted (this election will be the first I participate in, if I can). I looked up this past voter registration, and the state's website says I have an inactive voter status. Annoyingly enough, I tried to download their voter registration cancellation form, but the link was broken. Should I be worried about registering to vote in two different states?

Edit: I live off campus in an apartment not affiliated with my college.