We start by enlargening the incompetence at Norfolk naval yard. CSA captures more stores and especially more ships completely intact. They are quickly fitted as raiders and make it out to sea before the Union can enforce a proper blockade. The Union blockade will be weaker since more vessels will be needed to chase CSA raiders. The CSA can get some cotton out and some guns and powder in.
CSA promises Frane to support Maximilian. While Napoleon III does not want to enter the war without Britain by his side (and the Brits will not go to war to support a slave nation), he allows the CSA some credits and discount purchases of arms, especially artillery. Much is smuggled from French-held ports in Mexico to Texas and Louisiana.
Farragut steams into fire in April 1862, and USS Hartford is hit in its powder magazine, exploding and sinking with nearly all hands lost, including Farragut. The attempt to capture New Orleans is a failure. CSA retains one of its largest cities and best ports. The lack of support from the south and a substantial navy on the Mississippi will delay the Vicksburg campaign a lot. The addition of with French arms well-armed Texans and Lousianians together with the industries (as they are) of New Orleans and Jackson (which is not destroyed as early) improves the CSA ability to fight in the west.
The economy is much better (although still bad) because of the later tightening of the blockade and French credits and New Orleans still being in CSA hands. The hyper-inflation of the CSA dollar does not start off until late 1863, and is not as bad as OTL.
With less USA ships (since some are losr at New Orleans and several have to chase raiders), more Confederate ships (having New Orleans and less Union ability to attack ports and harbour forts), the CSA coast is less harrassed and thus needs less garrison and produces more for CSA.
USA now faces a grueling slow land campaign on three fronts - Virginia, Kentucky and down the Mississippi. It might take them until 1867 or even 1868 before they finally force the last CSA army to surrender.