The Union
The Republicans were not a single issue party. They were the party of the farmer, of western expansion, of the transcontinental railroad, and of industrial expansion. OTOH, they are a recently formed coalition of former Whigs, Know Nothings, Free Soilers, pro-War Democrats, and Democrats disgusted with southern dominance. Poor leadership or successful attempts by the Democrats to woo back voters could erode their voter base, but they’re unlikely to implode.
The Democrats have lost half their voters to the Confederacy and some of the rest to the Republicans. Since the demise of the Federalists they’ve been in the driver’s seat, but they aren’t any more. They’ve alienated both the pro-War Democrats and the Breckinridge faction Democrats, attempting to woo one of those groups back pretty much guarantees they won’t get the other one back.
Several issues have come out of the war. There are a large number of pro-Union southerners, black and white, in Union Army and they can’t exactly go back. There are a lot of black Union soldiers that have been enslaved by the Confederacy. Large sections of former Confederate territory are occupied by the Union – many but not all of these people are Unionists. Other Unionists still live in Confederate territory, some are in armed rebellion against the Confederacy. And considering the economy of the Confederacy, there will be economic refugees from the there.
The Confederacy
The Southern Democrats were a single issue party - slavery. Short term, they have factions, not political parties. Men sharing views on one issue may disagree on others. And they are riven with personal conflicts even among men who share the same views.
I expect the election of 1867 to be one of the messiest in history. All of the issues listed above will be brought up as well as expansion, inflation, revanchism, the public debt, and the size of the military. I’d expect each faction to field at least one candidate, there might be as many as a dozen men running. Some possibilities Atchison (Fire Eater), Rhett (Fire Eater), Ruffin (Fire Eater), Wigfall (Fire Eater), Brown (Moderate), Graham (Moderate), Hunter (Moderate), Stephens (Moderate), Vance (Moderate/Reconstructionist), Breckinridge (Nationalist), Thomas Bragg (Nationalist), Cobb (Nationalist), Harris (Nationalist), John Reagan (Nationalist), and Toombs (Nationalist).
The Confederacy may not survive this election.
Bad as things might get I see little possibility of a military coup against the Confederate government. The CSA government is broke. To cut expenses, I expect them to demobilize most of the army, leaving that burden on the state militias. If the Confederacy doesn’t send most of those troops they’ll face famine (if those troops aren’t sent home) or mass desertion (if they try to keep the troops in arms).
The Confederate Army was just as riven with factionalism as their civilian government. The only man with enough support from the CS Army that he might be able to seize power was Robert E Lee and he would never do that.
OTOH, I expect conditions to be bad enough in an independent Confederacy that at least one state militia seizes power in an individual state. Which creates another political dilemma that could destroy the Confederacy. Does the Confederacy accept this or do they send troops in to restore the civilian government of the individual state?
Another consideration is Trans-Mississippi Department, which may be physically separated from the rest of the Confederacy due to Union control of the Mississippi River. Kirby Smithdom becoming a military dictatorship is a distinct possibility,