In PA (and I would think the same applies in TX) under the circumstances you describe the other driver is liable to pay for the total fair market value of your vehicle at the time of the accident. The other driver's insurance company is responsible to indemnify their insured up to the limits of the coverage the driver had in place at the time. If that limit is at least as high as the value of your vehicle then you should get paid and promptly.
I'm concerned for you, however, because of the apparent delay. If the other driver admitted fault and the other driver's insurance carrier has already sent a rep out to inspect the vehicle then getting payment should take very little additional time. Their evasive response to you, suggests to me that they may be contemplating an offer other than 100% of the value of your vehicle.
You may want to consider turning this into your own insurance company, particularly if you had collision coverage. If you delay in notifying your carrier and that delay prejudices your insurer's ability to investigate, the delay could jeapordize your coverage.Ask a similar question
Sign up to receive a 3-part series of useful information and advice about personal injury law.