If you're using any of the crossings in or around NYC, the New York E-ZPass is your best bet because the savings will likely offset any discounts you forego at other toll facilities outside NY.

As for express lanes, the operation varies by state/area. The express lanes south and west of DC (all in Virginia currently) are HOT lanes, which means the lanes are free to use if you have 3+ people in the vehicle AND you have an E-ZPass Flex (which has a carpool mode). In contrast, the express lanes north of Baltimore have no passenger exemption - all users pay a toll. As such, it makes more sense to look at the Virginia HOT lanes.

If you don't have more than three occupants in the vehicle, then it makes no difference whose E-ZPass you have (you'll pay the same toll). If you have 3+ occupants but you are traveling when traffic is light (and the lanes happen to be open in your direction), the carpool mode might not save you enough to offset the higher tolls you would pay in NY with an E-ZPass from another state (historical rates are no longer available, but I think you'd look at under $10 on something like a Sunday morning). On a Friday afternoon heading out of DC, however, you might come out ahead with an E-ZPass Flex from Virginia (the toll at the current moment from the Beltway to the end of the lanes is nearly $30).

It's a little more inconvenient, but this might be a time when carrying two E-ZPasses makes sense (use the NY tag as your primary and carry the VA Flex transponder in a read-proof bag or wrapped in foil). That way you have options when you approach DC. Virginia does not charge fees but has an inactivity rule, so using their E-ZPass once in a while would keep the account open.