I have a non-CS major account (tmpclass) right now, and I'll be eligible for a CS undergrad account (under) soon. Can I just get my current account changed to an under account?

Yes, you can have your current account upgraded. Contact help@cs.utexas.edu after the beginning of the semester in which you become a CS major -- we don't get registration information until then -- and request your account be changed to an under account.