Unless there are other circumstances, you might have a really hard time moving unless your ex agrees to the move. The point of that clause is to keep both parents close so each can have close access to the children. Imagine how YOU would respond if HE had custody and wanted to move the kids... Probably not a good situation.
If your ex fights the move, your options are limited... 1) you could let him have custody. The moving clause is really only for the kids, you can realistically move wherever you want, but the kids stay here. 2) make a deal with your ex so he agrees to the move.
Obviously, if you are moving, you should offer to pay for all transportation costs related to the now long distance visitation. You should also read up on what long distance visitation means in TX since it could mean you are without your kids for 42 days in the summer. The visitations are less frequent, but much longer.
The other problem with your scenario is it's really your husband who has been offered another job out of town... not you. Have you considered getting a job so your kids can remain near their father? That would be the right thing for the kids... not moving them to Austin and away from their dad.
He certainly doesn't and should never offer to pay your bills so you can remain here, especially since you aren't working either. If $600/month is appropriate for his income, then there's not much you can do there either. The TX Child Support calculation is pretty straight forward. Not much room to fudge numbers...
Sorry, but the law is on his side here, just like it would be on yours if the situation were reversed.