As Mark's answer indicates, you are evidently thinking of the Full Faith and Credit Clause. "Public acts" being laws, it may seem at first glance that states must fully respect the laws of other states. But the interpretation of this clause by the courts is rather different, and has evolved a bit over time.
The short of the (modern) matter is that it mostly applies to matters concerning the judiciary. SCOTUS has recognized a "public policy exception" to the clause, which limits the ability of the clause to force a state to abide by laws which are in conflict with their own (for the most part: they don't have to). Driving privileges, and more generally who is licensed to do what (doctors, hunting, concealed carry, etc.), within a state falls under that public policy exception. So Texas does not have to obey New Hampshire's laws concerning the legal privilege to drive.
As a basic sanity test, if this were not the case, then why wouldn't everybody in Texas not simply bounce off to New Hampshire for a summer to get their license there and then return to Texas and never bother with insurance? It entirely undercuts the state's sovereignty and ability to set their own laws if any other state can so easily create loopholes around them.
Moreover, despite what the name might suggest, a "driver's license" is more a certification that you have the requisite skills, physical performance (passing an eye test), and knowledge to drive safely and in accordance with that state's traffic laws. It certainly makes sense for a state to require you to demonstrate at least that much, but they may also impose additional requirements. A requirement for insurance demonstrates your ability to handle financial liabilities that may reasonably result from your driving.
All states currently accept a valid out-of-state license in the above sense: that you are certified to have the requisite skills, that it is valid proof of age, etc. Though if you become a permanent resident there they may require you to take new tests. However to legally drive in any particular state you must not only have such certification (a driver's license) but also satisfy any other conditions, such as age requirements and insurance requirements.
As an aside, such state-by-state variations as to who is licensed to do what are in fact quite common, especially across history, even on very prominent issues. But even nationwide resolutions of those issues via SCOTUS have never, to my knowledge, utilized the Full Faith and Credit clause to do so. And, really, how could they? By saying since some state could force all other states to do X via the clause, then X must be a constitutional requirement? Or that any one state could unilaterally dictate laws in all other states? Madness!
For one example, anti-miscegenation laws, which outlawed (certain) interracial marriages, were quite common until 1967, when SCOTUS struck them all down using the 14th amendment. More recently, gay marriage was forced to be recognized in all states, also via the 14th amendment. In both cases, before those SCOTUS rulings, the courts had generally recognized that the Full Faith and Credit clause did not compel the state to recognize (out-of-state) marriages it did not want to recognize. These both fell under the public policy exception.