I called a local gun shop and asked them the questions you had.
Here's what you need to do and how long it will take.
1st, you will need to have receipts of at least 3 months showing you have been in Texas for at least 3 months, ie: phone bill, electric bill, etc.
2nd, you don't need a TDL. As long as you have permanent change of status from the Military, stating you are being transferred to Texas;
3rd, current Military photo ID.
Hope this helps, and good luck.
Since it's just leave, you may not be able to purchase while you are here. You may have to check a little further, but it sounds to me if you are stationed in Hawaii, that the gvt considers that your permanent residency until you are reclassified to another locaton.
HL