Instead of agreeing to Grant's surrender, Lee follows one of his staff's suggestions and disbands the army, telling them to head for the hills and continue a guerrilla campaign against the Union. Other generals soon follow suit. Lee is hanged for his continuation of the war.
Lincoln is shot later in 1865, Johnson hardlines, calling for more soldiers from the states and expanding the draft. More draft riots, more draft dodgers. In 1868, Johnson is unpopular, the death toll is rising,
Grant is elected with a 51% majority, being the only general who seemed to have gotten anything done. He loses 1872 to a Peace Democrat who recognizes the Second Republic of Texas and commands all Union troops to leave South Carolina saying that "South Carolina hasn't wanted the Union since 1833, let them leave and bring their slavery with them. All Negroes Lincoln has set free, soldiers will escort out. All Confederates who want to leave, go to South Carolina."
The secessionists and veterans who will not accept compromise go to South Carolina or Texas. Those that stay in the other states eventually compromise for peace. Louisiana especially is given certain autonomy after the years suffered under "Beast" Butler and its vital importance the the Union and southern economy. South Carolina and Texas eventually have strong economic and military ties. South Carolina, however, looks at Britain for its future economic success... only to find out that Britain is getting the majority of its cotton from India and Egypt.
German immigration makes Texas abolish slavery by 1900. Free labor makes Texas a rich nation. Oil fields make Texas a rich nation, polarized and often one-sided politics makes Texas a very conservative nation. Texas and Pacific Railway is eventually completed in 1930, making Texas more industrial and populous, making the country very very rich, eventually growing apart from South Carolina which is growing closer and closer to totalitarianism.
Speaking of South Carolina, their cotton-based economy fails to abolish slavery until the 1990s. Much like Texas, their politics remain extremely conservative. South Carolinians discriminate not only against Negroes who were unable to leave the country, but Confederates who emigrated instead of staying in the US. The Union largely ignores South Carolina. Occasionally, charitable organizations offer to cross the heavily-guarded border (the Charleston government is extremely paranoid for a renewed Union attack), to give food and medicine to the growing underclass in South Carolina.
The German government during the Great War sends two telegrams, one to Mexico offering them to regain lands they lost in 1847, and one to South Carolina, offering them weapons to restore the Confederacy. Once the United States discovers the telegrams (assuming they do) the Union and Texas join the Allied Powers and both send troops to France.
The incident leads to stronger relationships (especially with the oil) between the Union and Texas. Occasionally Texan and Union politicians begin thinking about reunion, perhaps achieved before the 21st Century. South Carolina remains almost a Latin American style dictatorship, facing dangerous urban environments and vigilante justice in the countryside.