I'm not sure about quals, I just graduated prototype, and I'm yet to go to the fleet. Based off the orders my class received, if you're looking for a ship in drydock, it would be very easy to get sent to one. I'm new to OPSEC and all, but there is a ship in drydock in norfolk for about the next 5 years, and a good portion of my mechanic friends got dumped on it.
As someone with a family member in a possibly similar situation, I'd recommend going where you can benefit the most. You won't be able to there for them as much, but I know if it were my family, they would understand and wouldn't want me to hold back my career. If things get really bad, emergency leave is a thing, and as long of your chain of command is aware of what's going on, they can help out a lot.
It will definitely limit you with being qualified for chief, though (correct me if I'm wrong) you obviously won't be able to qualify LPO at sea, and from what I've heard, qualifying EWS on your first sea tour requires a lot of hard work and luck. That might not be terrible though, as from what I see employers in the civillian world don't just look for quals, they want to see you standing those watchstations (for at least 2 yrs?), so one sea duty where you were qualified EWS for 2 months before leaving the command isn't necessarily helpful.
If you want to get all the quals that make you competitive in the civillian world, you're already looking at multiple sea tours/prototype instructor tours, so your first sea tour being in drydock wouldn't necessarily be a bad move. Again, if any of this seems wrong, I'm sure the more experienced posters on here will correct me, and thank you in advance.
Best of luck, boat partner.