I believe if you file in New Hampshire, the state will have jurisdiction over your case regardless of where the other spouse is living. I think this is true even if the marriage was performed in the Philippines but I'm not sure. You can probably either file a "fault" divorce for abandonment or go for a "no-fault" divorce (basically claiming "irreconcilable differences"). No-fault is usually simpler since you don't have to prove any grounds.
Unless YOUR religious beliefs require it, I'd ignore her request for an annulment. If she wants one she can take it up with her priest, but you shouldn't be responsible for paying for it.
You should probably consult with a local family law attorney about how to do a divorce if you think your ex will be uncooperative or nonresponsive. I don't think there's any way she can force you to stay legally married if you want out, and the lawyer should be able to tell you the best way to proceed.
Edited by TJMH (04/13/16 05:20 PM)