I realize this is about a month late, however, there wasn't really a good answer to the OP's question.
1) Generally speaking, when SWAP is instituted there have already been other traffic management initiatives in place i.e. miles in trail, minutes in trail (not widely used), airspace flow programs/flow control areas, etc. to manage the demand heading into the NY area.
2) There are three different sets of SWAP routes: SERMN (SWAP Escape Routes Metro NY) North, South, and East. They can be implemented separately, together, or in any combination as needed. Usually, you file normal routes and request or are assigned the re-route. You must fly the route exactly as depicted and you will be altitude restricted until leaving ZNY's airspace. My advice is to make it known to ATC you are willing to accept a SWAP routing. In addition, being a pathfinder aircraft could help your cause as well.
3) The parallel off-shore, deep water routes you could be talking about are specifically referred to as the AZEZU routes. They are available depending on whether or not the military is willing to provide some VACAPES airspace for civilian use. They are meant as an alternative option for those operators flying between the Northeast and Florida or vice versa. Excellent option if they are available and you are equipped.