You probably need a lawyer's help, but look at rule 201 of the Texas Rules of Civil Procedure. It says how to take a deposition for use in out-of-state proceedings and that sounds like what you need.
This is not legal advice. You should always discuss the specifics of your issue in person with an attorney. Be aware that there are time limits on all claims that depend on the kind of claim, so do not delay in seeking an attorney.
Richard's answer above is correct, but you should also consider the options under Rule 202 (see link below) to use Texas Courts.
Under that Rule,you can subpoena records and get a order for an actual in person deposition if you need that much. You can do that with a video tape instead of a costly Court Reporter. The fees for the Court would be around $300, but if you can get the local firm's attorney to waive the required deadlines and do an agreed or unopposed filing, it will go much faster. Otherwise it will take 15-30 days minimum. Also, if you are really in the other out of state lawsuit and subject to THAT court's jurisdiction (which may not be the case) you should be able to get a subpoena or use the other State's Court's rules to help you. I only am licensed in Texas State Court and certain Federal Courts.
You really do need a Lawyer FAST.
If you want a short no fee, no obligation this afternoon or weekend, I can try to fit it in.... bring all the papers. Just email me to set it up. EK@bizlaw.pro