Can you ask judge for a certain date if you have proof that will be the date you are moving
If the date is reasonable of course.

You can ask, he can agree or he can order you out at once,
You should be ready to pay at the court date in full all money up to that date you want to stay. If you do not and can not the judge may order you removed at once.

I suppose you can ask for whatever you want.

In Texas the standard procedure is for you to have 5 - 7 days from the date of the hearing, once the ruling for eviction has been made. Then if you're not out by that time the landlord has to go file for a writ of possession. The sherrif or constable will come give you a final notice that they'll be back the next day (or so) to excort you off the property while the landlord changes the locks and has your belongings removed.

If you plan to be out within a week of the hearing you're probably in good shape. If it's much longer than that I'd start making alternate arrangements... or pay your rent.