That's not how unemployment works, at least not in Texas. You wouldn't go off the unemployment until you were actually hired. You would simply report the income during that period (2 week periods in TX) and they deduct the amount from your benefits for that period.
Yes, most states encourage you to take work, even temporary work. A contract job won't require you to stop it and start it, you just "skip" a week.
Last time I was on UI, it didn't actually last for X weeks, it lasted for Y dollars, and if you earned more than a certain threshold in a given week, they would reduce your payout that week by 50% of the amount you went over that threshold. So one week of high earnings would, in effect, mean you get $0 that week but give you an "extra" week at the end.
