Rex, what I find amazing on this forum is how little has changed since I was in the Navy (1966-72). Nukes are Nukes. Most complain about the conditions, but most also will admit it is was an experience they are proud to have.
Anyway, I feel your biggiest challenge is your age and your family. You are 24 going on 25, when I was 23, I was considered an old man, but had the rank (Nuc ET, E5 in two years, E6 in four). But worse then that, there is nothing in your experience or your family's that will prepare them for where you are going. Things won't be to bad for the first 1.5 years, then it starts. Prototype training was a seven day rotating 12 hour shift when I attended the unit a Balston Spa, I assume it still is. Even this is easy on the family until you reach the fleet.
Having worked as an RO in engineering and a lead ET in operations, I can tell you with absolute certainty you are going into the worse surface fleet duty sector the Navy has to offer. While it is true, the sea duty will be the same, it is the in port duty that is rotten. Most divisions have in port duty sections of around 5 days or so, engineering does not. Unlike the radio guys, or the weapons guys, even a cold iron reactor has a watch. If you are hot, the engine room has to be manned at a higher level, and a critical reactor is manned at the same level whether at sea or dock side. So look to lots of 3 section duty (one off, two on) if you are lucky.
Life in most parts of the Navy is a life apart from your family, and that can, and has been a hard experience for many.
It is an adventure, but it is not glamorous, and there is nothing wrong with going for the money, just realize it is a salary. I remember an E5 RO when asked by the watch officer something he did not know reply, "Sir, what do you want for 50 cents an hour, an Einstein" Yep, back then that's what it worked out to be.
As far as ET, EM, or MM, after the service, it does not matter much. I got out and became an accountant, then retired as an IT manager. You never know how it will turn out.
Good luck to you and the family
Noelan