JALLEN brings up a good point. If you change your DL or address during this process, it's a real pain.
Fortunately, having your Texas DL helps make this easier...
Since you haven't resided in Texas for at least 6 months, you fall under the GC §411.173 "Not quite a Texan, yet!", or Non-Resident category.
If you wait to apply until after 6 months, you will fall into the GC §411.172 "Whoop! I'm a Texan now for 6 months!", or Resident category.
Either way, you can take your LTC class today, and then either wait until November to apply or not.
Assuming you meet all the other requirements, you can apply immediately OR in November! The choice is yours!
Note that your application WILL ask your previous residence(s) for the past 5 years, and since you were out-of-state during that time you might have to pay some additional fee for running a background check in Illinois. In fact, GC §411.173 code specifically mentions "...payment of a fee in an amount sufficient to recover the average cost to the department of obtaining a criminal history record check and investigation on a nonresident applicant." Also, note that this fee is not specifically mentioned in GC §411.172. So, might be cheaper to wait, might not.
Anyway, Good Luck, and welcome to Texas!