How is the vehicle titled, in her name? If it's in both names, needs to have both listed with *OR* in between them. Depends on the state (not familiar with Texas), often there's some legal mumbo jumbo / estate stuff that needs to happen before the title can be transferred if it's in both names with AND between them. Maybe you have that figured out already but figured I'd mention it.
Regarding the loan, private party sales happen all the time when there's a loan on the vehicle. Typically the process works as follows:
buyer & seller reach an agreement in writing
buyer writes check / hands cash, or other verifiable payment for vehicle
seller sends money to bank and demands a lien release and title (if a non-title holding state)
once the lien release is in hand, buyer and/or seller can take the title to be transferred.
There's a significant delay in all the above which is why so many people do dealer transactions when financing is involved. When to hand the vehicle over is a sticky subject, if you trust the person and you have guaranteed money in hand, sometimes sellers will let buyer take possession of said vehicle (with verification of insurance) before title work is complete. To err on the side of caution, wait till you have the lien release and can finalize everything before handing the vehicle over. How you approach it (when it comes to liability) depends on what state you live in.