It might be possible to apply for a license in another state but 40 plus states are members of the Drivers License Compact (DLC) which requires members to share driving record information. Since most states exchange information regarding your motor vehicle driving record, these states will check with Texas to see if your TX license is valid. Therefore, you won't be able to get a license in those states until you have your Texas license reinstated.
For example if you had your license suspended in another state and were trying to obtain a Texas license, the Department of Public Safety asserts that a person who has a suspended driving status in another state is not eligible for the issuance of a TX license. To obtain a TX license due to an adverse driving status in another state, the driver must obtain a clear status from that state's driver licensing agency.
How Texas deals with this situation is how many states also deal with it. Even if the state is not part of the DLC it can query the National Driver Register (NDR) to determine if an individual's license has been withdrawn (suspended, revoked, cancelled or denied) by any other state. All 50 states and DC participate in the NDR.
If there is a certain state that you are interested in moving to, you can check with their department of motor vehicles to find out what that specific state's guidelines are regarding out of state license suspensions and obtaining a license in that state.