Wait a minute. She had an affair. Now you must leave the house. And you must leave your kids. And you must leave the state. All of this to prove yourself to her? Sounds like she is pulling one over on you to get you out of the picture, not only with her but the kids too. You dont need to leave, she just wants you to.
If you leave VA without a costody order and filing she will 1) prevent you from returning, 2) move another man into the house, and 3) hit you with child support and possibly spousal support which will be deducted from any job you can get in NY.
Even if you find great success in NY sorting our your life and getting a good job, why in this situation would she ever want you back?
My advice to you would be to do everything you can to stay in the house as long as possible, do everthing you can to find any job, and spend and document as much time with the kids as you can. If you must leave, make sure you are legally protected first.