In general, a court doesn't grant a stay of execution unless the tenant can pay all current and delinquent rent to the court when filing the request. You could run the risk if you pay only a portion of it, that the court would deny the stay and then turn over the amount paid to the landlord if he has gotten a money judgment as well. Then you would be evicted as well as have lost any money you could use to move.
There is no guarantee it will be granted, but it might be better to file without paying and also request a hearing. Then at the hearing you can see if you can negotiate a payment agreement by paying a lump sum and working out a plan for the rest of the money with the judge's help.
If you really want to stay in this apartment and you will have the money on Weds, it is possible a judge would consider a stay for a few days in order to let you get the rent deposited.
If you are able to consult a landlord-tenant attorney, I would advise that to have an an advantage in court by using a lawyer IF your hearing is granted. In the meantime, since you are running out of time, there are forms you can get online from private companies and probably also from the TX court website that can be used for a motion for a stay of possession.
There's not enough time to find an attorney for this but you may be able to call the court clerk and see if they can direct you to the form for a motion. If not, follow the court website or use a private company (such as Rocket Lawyer) to create a motion for a stay in TX. You can always try getting a lawyer if you are granted a hearing. If you need help getting a landlord-tenant attorney, I'd just need to know the name of the biggest town or city in your area and I'd be happy to find a landlord-tenant attorney for you if you get a chance to have a hearing.
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