The problem with this scenario is, as other posters have pointed out, for the string of victories to keep going requires magic. Let's take Washington DC for example and say, hypothetically, Lee is at Washington's gates by the time of Antietam OTL. He'd be facing what was, at that point in time, one of THE most heavily fortified cities on the planet and nowhere near enough siege artillery or other means to really hope to scratch the ring of fortresses surrounding the Capitol. It would take the commander of the defenses committing an act of borderline treasonous stupidity to lose in that kind of a situation even before you take into account the likely Union reinforcements en route who will arrive before any Confederates do.
How long would the munitions and food of Washington last at the time of Antietam? Of Manassas?
So July 1861, after Manassas...
Washington is on a peninsula. Yes, Washington is fortified and hosts concentration of Union forces. And Potomac is wide below Washington. So how about crossing Potomac north of Washington?
It is not "treasonous stupidity" for Lincoln and Cabinet to stay in their capital and lead defence in a reasonable hope that the Rebels will be repelled. Reasonable but, in the event, incorrect hope. It is also not "treasonous stupidity" for Lincoln and Cabinet to try and run while they yet can, instead of sitting under siege with no contact with Union main forces and eventual capture after months of hunger. Yet when the hope to repel the Rebels turns incorrect, the hope to get through by train or foot may, although reasonable, prove incorrect.
As for the reliefs - yes, they are coming. Like Pennsylvania Reserves.
But again. It is not a "treasonable stupidity" to attempt to relieve the besieged capital as soon as possible. It is rather action in desperate hope. Which may prove incorrect.
Assume that in August 1861, several Union forces march to try and recover Maryland, and relieve Washington and/or other garrisons that may be under siege, like Baltimore. The Confederates are still waiting for siege guns - but the besieged Union forces are eating through their food and munitions. Several Union forces, precisely because of the desperate urgency individually small and ill led, are defeated in several battles in northern Maryland and southern Pennsylvania.
What will the northern governors do?