It's hard to say. Without Texas and Cailifornia you don't need the Compromise of 1850 - yet - but it's a patchjob that satisified no one. For example most people in the non-slaveholding states could have cared less about slavery until the Fugitive Slave Act started trampling all over their States Rights.
But without Texas, there will only be one route for the transcontinental railroad - the north. The central route used in OTL and the southern route that led to the Gadsden Purchase just won't be available. This would lead to southern complaints that the plan unfairly favors the north. (Of course, they were perfectly fine with government funds being spent on the Gadsden Purchase.)
And there'll be no additional slaveholding states unless they attack Texas and/or expand into the Caribbean. I'd see greater southern interest in an expanded navy, while the rest of the country doesn't want to get dragged into foreign ventures.
Important questions are when the Whigs/Know Nothings/Republicans/whoever manage to put together an effective not-Democrat Party and when the rest of the Democratic Party gets tired of the southern branch of the party calling the shots.
In OTL, both those happened at about the same time and led to the south starting the ACW. With an independant Texas, that could happen about the same time it did in OTL, earlier, or later. And they might happen far enough apart there is no ACW.
Of course, an independant Texas is going to have a rocky road ahead of it. The southern USA/CSA and Mexico are both going to want it and both have significantly larger populations to draw upon if it comes to war.