Renting in Devon and Cornwall is NOT more expensive, especially in Plymouth, if you're sensible about where you choose to live. Have you looked at Torpoint for instance, or at Saltash? Good primaries there, and rentals available on the new Pilmere estate. You could also look at Manadon Park, and some of the roads off Peverell Park Road always seem to have properties for rent, and on the Pennycross side as well. There's a 3 bed cottage in Millbrook, Torpoint going for £495 per month. I've just looked on the net.
You have had the time to sort this out, and have chosen not to. Why haven't you been making the running on this when he's been at sea? How long since he put his letter in? My understanding is that takes a year to 18 months to leave, so you've known it was coming.
It is perfectly simple really. You look at your options, including housing, as the rent will be somewhat different from an MQ; he decides to put his letter in and leave the RN, and you start making enquiries about housing, sooner rather than later, moving out of the MQ before he leaves the RN if you've found a rental you like. You are not being evicted fgs, it's effectively tied housing, he no longer works for the RN, no entitlement. When dh leaves his current posting, it'll be the same for me, so I shall march out of this MQ, and go back to the UK, having ensured that I have somewhere to live when I get back there; either another MQ, or my own house, if I can give the tenants enough notice.
Your dh may have been in the RN for 10 years, but you haven't, so you have no excuse not to have got this moving. My dh has been in for 30 years, having joined at 18, and we have a family too, but we've always fended for ourselves. You need to plan ahead. My dh leaves the RN at 53, but we're already looking at options for when he leaves and putting out feelers for possible jobs. Prior planning prevents and all that.
People coming out of the services DO get help if they want it, there's loads of info on the RN website with links.
I really don't see how your situation is any different from anyone else leaving a job and having to rearrange accommodation. Yes, there is a measure of protection in Naval life, and it can be a bit detached from reality for those in the RN, but you, like me, are a civilian and live in the real world, so for you there should be no transition to make, apart from having your dh around a bit more.
Word of advice, if you do rent privately, avoid Executive Lets on Mutley like the plague. You may also find this useful
www.accessplace.com which has a list of the HAs in Plymouth, with addresses, phone numbers and websites.