Say you have the scenario Dave Howery proposes: that Travis and Crockett abandon the Alamo in favor of reinforcing Goliad. The battle at Goliad might be more successful than OTL, perhaps enough to allow Travis and Crockett to escape a likely, though more costly defeat. There are still a number of questions: how does Santa Anna respond? what happens in San Antonio de Bexar?
For one, it's important to recall that the force at the Alamo was originally the Army of Volunteers, whereas Travis was a commander of the Regular Army; he and Bowie were in joint command at the Alamo, each of their respective forces. Bowie's illness was thus lucky because it left Travis in sole command of the garrison. If Travis and Crockett survive, the latter may well do so as commander of the Army of Volunteers. Both may come to resent Sam Houston for not relieving them either at the Alamo or at Goliad.
IMO, it's still likely that Santa Anna will attempt to press on after a victory at Goliad. If so, then it will probably be Sam Houston's force that meets and potentially defeats him. However, if the Battle at Goliad is more successfull, perhaps Travis and Crockett manage to negotiate a settlement. I doubt they are able, because it will take a bit longer for the political situation both among Santa Anna's commanders and back in Mexico to progress to the point of overthrowing him.
Without spending too much time on specifics, it looks the politics of the early Republic will be pretty interesting. For one, say Houston and Crockett both enter the race to be the first elected President of Texas; however, they end up splittig the vote and Stephen Austin wins. Without OTL's Battle of San Jacinto, however, Mirabeau Lamar is not likely to be Vice-President. If Austin is successfull as President, I kind of like Juan Seguin as Austin's VP and eventual successor, which will inject Tejanos a bit more forcefully into the Republic.
A split between Crockett and Houston might well fester: emnity from the divisions between the Army of the Volunteers and the Regulars reinforced by anti and pro Jacksonianism respectively. Without Lamar, this split, mitigated by the Austin/Seguin crowd may define the Republic. If so, then these two factions will agree on a lot: beneficial treatment of Indians, for one.
The long term prospects of the Republic of Texas will nevertheless depend upon 1) the ability of the Republic to attain foreign loans, 2) Mexican politics and any future invasion attempts, 3) US expansion/slavery politics. Only one of these--foreign loans--is really something that Texas might influence and even so is more likely to depend upon foreign powers than Texas' politics. However, if Texas manages to avoid the expense of the Lamar expeditions and Indian policies, attract Tejano sympathies particularly those of Santa Fe, and develops a more mature politics (Crockett vs Houston is not likely to develop into the kind of antics that characterized OTL, such as the Archives War, IMHO), then Texas may find itself more able to survive as an Independent country. However, it's really all about the money: Texas was always strapped for cash in a serious way.
Even if such a Texas were annexed by the US, it might create interesting US ripple effects: Houston and Crockett might continue their political careers in Washington, affecting the politics of the 1850s. Either could make for a nice compromise candidate for the Democrats convulsed by slavery and resentment of Stephen Douglas. Texas might retain a bit more Tejano influence if the patterns of settlement after annexation are different.
