The numbers are :
British took or destroyed 34 US warships, 278 privateers and 1,248 US merchantmen
USN took or destroyed 16 RN warships and 172 merchants; US privateers took 1,441 merchantmen. Of the merchants ca. 30% were immediately recaptured by the RN and never made a US port.
The American offensive against British shipping was ineffective, taking only about 5% of the entire UK merchant marine during the course of the war, and seeing a massive increase in British shipping due to captures of US shipping being greater than the loss rate and the reflagging of US ships to the UK.
In 1861 the French Navy is massively greater than the USN. Remember that privateering is out, the Treaty of Paris means all privateers will be hung as pirates rather than treated as PoWs. The USN needs to send regular warships and isn't likely to have more than it had OTL. The French as of ca. 1860 had:
9 screw ironclad batteries
14 fast screw battleships + 25 converted screw battleships
7 fast screw frigates + 12 converted screw frigates
10 fast screw corvettes + 2 converted
24 fast screw avisos (sloops) + 3 converted
28 screw gunboats and gunvessels
19 paddle frigates
9 paddle corvettes
50 paddle avisos
14 paddle tenders (general purchased ships carrying mortars)
13 sailing battleships (all effective and awaiting conversion to screw, but this was cancelled due to the armoured frigates)
40 sailing frigates (12 converted to screw before 1862)
18 sailing corvettes
They of course had added 6 armoured frigates and a few other ships.
The USN on 31st December 1862 has:
2 Ironclads (New Ironsides and Galena)
4 Large Screw Frigates
3 Small Screw Frigates (Corvettes in French terms)
3 Sidewheel Frigates
13 Steam Sloops (no attempt to separate out any sidewheelers)
3 Sail Frigates
11 Sail Sloops
4 Sail Brigs
30 Transports
9 Schooners
19 Mortar Schooners
3 Barks
2 Yachts
8 River Steamers
4 Ocean Steamers
40 Gunboats
47 Auxiliary Gunboats
58 Transport Steamers altered to Gunboats
The USN is massively outclassed by the French Navy. It's difficult to see any US steamer operating away from the US as they won't be able to resupply.