In short it continued to be used for another half century almost exclusively, and was even treated as a resource critical to national security that needed to be husbanded and regulated, but by the end had been deeply depleted for numerous reasons.

We should first state the answer to your last two questions. It was very much desired for its strength and longevity for use as framing timber in ships by anyone who could get it, military, and civilian. European naval and maritime interests quickly noted the wood's benefits and even in the 1600's its difficulty in cutting and working was still a good trade to get it for use in ships by the British, Spanish, and French as they could. But by the early 1800's the Southern and Gulf Coasts of the US held virtually all that was realistically available for ship building. Doubly so following the Louisiana Purchase and the 1819 Adams-Onis Treaty which transferred Florida to the US, and was already growing scarce by the time the Dept of the Navy was founded in 1798. Other nations turned to either trying to buy up what they could, smuggling, or using other woods they found to work well, the Royal Navy especially turned to Teak sourced from India and Malaysia(though /u/thefourthmaninaboat can speak far better on that).

In the mid 1820's as illegal harvesting and timber pirates on public lands continued to encroach surveys were undertaken of the entire coastal stretch of the South to establish the location and extent of useful plots of the wood. Land reclamation for agricultural, industrial, or residential usage also cut into its habitat, the assorted Live Oak types and closely related species only really grow in a 10-20mi stretch inland from the tide line.

The decade after the end of the war also saw specific plots of land bought up for reserving timber for the Navy(a single 44 gun 'super' frigate like Constitution might need 500 trees just for its framing)! And laws passed to further discourage illegal logging and theft, while the Navy Dept even laid out funds for small craft and vessels to patrol near their stands to protect them.

The last though proved relatively ineffective in rooting out the bad actors in what was a very large legal export and sea based trade. See this quote from an 1832 Report to Congress by the Secretary of the Navy:

Of the three small vessels heretofore employed in the protection of our live oak, there seems now to be little doubt but one unfortunately foundered in a severe gale during the previous year. The other two, under a belief that their further employment would not be so beneficial as to justify the expenses they occasion, have been recalled. One ceased to be in commission in May last, and was ordered to be sold, and the same disposition will be made of the other on her arrival at Norfolk.

Part of the saving grace though was that once harvested, if kept in proper seasoning conditions, the frames could last for years if cared for(occasionally this could even be stored underwater in a low oxygen pool or pond as was used to source timber for the Constitution's two 20th century rebuilds). Helpful in an era where the Navy did experience a post war boost in funding and spurts of newly authorized construction, but which ebbed and flowed, so ships often sat partially assembled, as built but unequipped hulks and receiving ships, or simply as theoretical assemblies of parts at a navy yard. But new builds, and simple maintenance meant the Navy always needed a supply of the wood, along with other types like Fir, Cedar, Pine, and other Oaks, all with their uses or niche roles in shipbuilding. In particular Pensacola Bay and the Navy Yard there were in a hotbed source area and its was hopped make managing the supply easier, in fact even today there is the Naval Live Oak Plantation there rolled into the larger local National Parks land.

An 1831 report to Congress by SECNAV Woodbury who also submitted the one quoted above speaks to the changes in political winds that impacted Navy efforts to secure reliable sources of Live Oak:

The plantation of trees, commenced a few years since in Florida, was not deemed a proper subject of further attention by my immediate predecessor, more, however, it is presumed, from doubts about the legality and necessity of that particular measure, than about the utility of careful attention to either the growth or safety of our live oak generally. No expense in respect to the plantation has yet been authorized by me, except such as seemed indispensable to secure the benefits of previous expenditures. A more extended plan of agents and vessels for the discovery and preservation of such live oak as is now growing on the public lands, was devised last winter, dividing the seaboard from the St. Mary's to the Sabine into seven districts, with an agent in each, and stationing three small vessels at proper distances on the coast, to aid in the same object, and in the survey of the bays, navigable creeks, and rivers of that region.

But while forestry management, policing, and even intelligent storage could help, ships have a nasty habit of always slowly falling apart!

From the same 1831 report several vessels are noted that are in need of attention, and helping to make the case to Congress that it wouldnt do to let them sit!

The Java, on a minute survey, was found to be very defective; and, having been built of inferior materials, expensive repairs on her are not deemed judicious. As she bears the name of one of our trophies during the late war, it is recommended that an appropriation be made for purchasing timber to rebuild her, and another, for a similar reason, to rebuild the Cyane. A frame has heretofore been obtained for the Macedonian; and an appropriation to finish one of these vessels, whose names are so intimately associated with our naval glory, could be expended with much advantage the two ensuing years.

Realistically though between Naval needs, and civilian shipbuilding its likely that supplies would have been exhausted before the 1800's were out. But in the 1860's things changed rapidly. The viability and desirability of Ironclad warships came to the fore, from small coastal ones like the Monitor to the huge HMS Warrior meant that the wood behind the armor, so long as it could support the structure, became less critical in the ships performance in combat and seaworthiness. It also posed a problem when the location of most of the Live Oak in the US was suddenly no longer answering the USN, and most Civil War constructions had to be conducted with replacement woods like other types of Oak. Following the war while it still saw usage in repairs, and civilian merchant ships at great expense, the USN also built very few ships until the late 1880's when the age of the steel navy was well upon them.

Sources

Secretary Woodbury's 1832 Report to Congress

And the 1831 one

A wonderful little timeline of Live Oaks in govt usage from the University of Georgia's School of Forestry, Go Dawgs!

A Naturalist in Florida by Archie Carr, with some excellent descriptions of the Florida plantations for Live Oak and the difficulties and character of those men sourcing it a portion of which is available online through Google.