Certainly an interesting notion, if a <<sniff>> depressing one.
Houston's early death could well have some sort of effect on Jackson, but just what is hard to judge. Houston was a Congressman in 1824 and helped Jackson campaign in the Presidential contest that year. His efforts failed, but the fact that he played a role means you'd have to judge whether it would have an impact on the events of the election of 1824--that election being chock full of very contingent, personality driven events and its outcome being crucial to the next 30+ years of American politics.
Houston met Alexis de Tocqueville in 1831 and is said to have formed the template for Tocqueville's portrait of a "nervous American." A small change, but Tocqueville is pretty influential.
The lack of Houston in Texas will be the most telling: first, he won't be around to help negotiate a treaty with the Cherokee in northern East Texas. This treaty was fairly important to ensure the Texians could face the Mexicans without worry, but there was another commissioner there, John Forbes. Whether he had Houston's background is far from certain. Next and perhaps most important is who becomes commander of the Texas army in February 1836. Houston's strategy was to withdraw into Eastern Texas, forcing Santa Anna to extend his lines and taking advantage of the more wooded territory which was better suited to Texian rifleman. Given the abysmal discipline of the Texan army, a less able commander might have proved unable to direct such a move (or prompted a movement to replace him by a direct vote of the troops). It could be all too easy to have events evolve in such a way that the Texans are defeated.
So too however, they could win in a different way and this might alter the status of peace to the Mexican government, which in turn affects the Republic of Texas' viability. Nevertheless, for the calculus of annexation (in terms of the benefit to Texas) to be substantially different, Texas would need European support far sooner (it's all about the public finances which were atrocious under the Republic).
Overall, it's an interesting notion, but precise speculation becomes very difficult: enough time elapses between the POD and major events moved by it, that one could always post that a person might take on a wholy analgous role (but didn't OTL because Houston did).