Yes, you would only pay tax in VA, unless you can prove that you left Virginia with no intention of returning. If you are able to do this, you may be able to file a nonresident or part-year Virginia resident return.
You are correct that TX does not have an income tax, therefore, allocating income is not an option. Being an intern is considered as having a temporary home in the other state, or in this case TX. As such, this income would still be considered taxable in your domiciled state of residence.
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