You may want to look at some of the following:
Pentagon Sailing Club. Their program can take you through the Navy Sailing and ASA curriculum up to D-CS/ASA 106 for a lot less money than most of the commercial schools. They run the lower level course on Capri/Catalina 22s on the Potomac. Their higher level courses are run on chartered boats out of Annapolis. DoD badge not required.
They award the B-CSN and ASA 104 concurrently. With the B-CSN, PSC has a strong relationship with Annapolis Bay Charters. You can then charter an ABC boat as a PSC member at the same discount rate over their publicized prices that they accord to PSC.
There are many deeply experienced sailors at Pentagon, who own their own boats on the Bay, and have extensive experience in the BVI. PSC runs an annual BVI trip that costs about $600 per person for 10 days of sailing. PSC skippers charter a lot on the Caribbean and are well-known there. Horizons will let PSC members charter by themselves with an ASA 104 or Navy Sailing B-CSN. However, PSC will not let you charter a PSC arranged "club event boat" unless you have a D-CS/ASA 106.
Navy Pax River Sailing Club. They have one of the best sailing grounds and boat setups in the Bay, across from Solomons Island. You will need a to be able to get a DoD badge, as they are at West Basin on the Navy Pax Air Station. They use Catalina 16s for ASA 101 and Catalina 250s for ASA 103. For their big boats (they use donated Naval Academy offshore boats..Luders 44 Yawls and Navy 44s), they switch to using the Navy Sailing Qualification process.
Both clubs will permit you to challenge to take out their boats at minimum cost.
Another good place to sail is Sail Solomon's which is run by Andy Batchelor and his wife. They offer the full range of ASA certifications and they also run a very economical chartering package.
J World has fabulous instructors. Their racing programs are really where they distinguish themselves. However, if you want to get a ecognized US Sailing, ASA, or Navy Sailing Credential, they will not be able to award it.
The Maryland School of Sailing in Rock Hall is the gold standard, especially if you want offshore experience.
The commercial schools are much more expensive than Pentagon if you are primarily looking for a 104 or 106 qualification. Go to JWorld if you want the racing, Maryland SoS Rock Hall for the offshore, and Sail Solomons if you want to charter bay boats cheaply.
If you are interested in learning about mechanical and electrical systems, the Annapolis School of Seamanship (John Martino) is outstanding.
As for distance, Rock Hall is a good three hour drive from Fairfax. PSC's river fleet is about 30 minutes from Fairfax in no traffic. Their bay boats are run out of Port Annapolis (about 1 hour 15 minutes from Fairfax). For Sail Solomons or Navy Pax, you are looking at about 1 hour 45 minutes of driving in no traffic.