If I understand correctly, Texas declared themselves their own nation, Mexico considered it Mexican, and the U.S. considered it American. So once the war was over, wouldn't Texas still believe itself a country? The only reason I can think of would be if the U.S. threatened to take Texas militarily unless they joined peacefully.
Texas won its independence in 1836. It fought additional wars with Mexico as it contested claims for land south of the Rio Grande river and tried to expand to the Pacific ocean. Texas also fought a major war against the Commanche Indians and had to suppress rebellions in three of its large eastern counties. Politicians in Texas were divided over the issue of annexation by the United States. Sam Houston thought it would be in Texas' best long term interest to join the United States, but his biggest political rival, Mirabeau Lamar, wanted Texas to remain an independent nation. The president of Texas was elected for a three year term in office. Sam Houston was president from 1835 to 1838 and again from 1841 to 1844. Houston won his first election before Mexico recognized Texas as an independent nation. Mirabeau Lamar was president of Texas from 1838 to 1841. Lamar was opposed to John Tyler's attempt to annex Texas, but it was primarily Northern states opposed to admitting another slave state that ham-strung the incorporation of Texas into the United States. When Sam Houston won his second term, public opinion in Texas swung towards union with the United States.
The Mexican American war was fought to fix the southern border of Texas and to allow the expansion of the United States to the Pacific Coast of California.
Source: "Lone Star: The Republic of Texas 1936-1846" by Stephen Hardin and Mary Dodson Wade
The United States annexed Texas prior to the Mexican-American war.