congratulation!!! From 350y it is just a small step to find your cruising "forever" stroke! (As you said yourself it's just a little bit more relaxation..) And you can't say anymore you fear putting your head under water... (starting with backstroke can help to get familar with the water and without face down... not for you!)
A diving mask produces much more resistance than goggles do. May be a nose clamb will help. If you get the right one an advantage of this is you can blow out through nose too, which can also help to more relaxation.
Enjoy with best regards,