Try contacting the society for trans people on
post@ftpn.no or phone number 93207921 (although I would probably go for email as there will be a lot of information going back and forth but you can always give them a call to ask if they can help at all), they have probably been through a situation like that before and should know a thing or two.
I would also contact the department of public health (in Cali) and ask them about the needed paperwork, not to mention continuing receiving benefits and how you should solve that practically. Regarding waiting until July or not I'd simply take into account both when the transition date is, along with how long you can wait (and should wait. Might be that the hospital also has some formal paperwork you need to go through prior to the surgery).
Have you considered applying for a Norwegian citizenship? After 8 years you should get it. Depending on how much you are receiving, becoming Norwegian and getting the Norwegian benefits from NAV instead might be just as good. Just a thought.
Anyways, good luck. I hope things turn out for the better.