I'm so sorry you're going through this.  Do you have Al-Anon meetings near you?  Longterm support is so helpful when we're going through the insanity of having addicted family members.

Of course the addicts/alcoholics cling to their "easy" lives and want to keep it easy and supportive of their addiction.  So unfortunately I'm not surprised that your sons are resisting. I think many of us here have been through a similar situation.

I'm guessing from your mention of the bedroom tax that you're in the UK.  If that's true, you might try this site to see if you qualify for legal aid:

https://www.gov.uk/check-legal-aid

This site suggests that legally speaking you can just give them a Notice to Quit and then change the locks:

http://www.justanswer.com/uk-property-law/4xuk9-homeowner-looking-evict-fellow-family-member.html#re.v/180/

(I hope it's okay to post this - Mods please change if not.)

HOWEVER, that's legally speaking, not emotionally and otherwise speaking.  Al-Anon does give us tools to set wise boundaries, which can mean being clear that someone has to leave and enforcing that boundary once it is set.  But setting boundaries is not always easy either on them or on us, and it's very helpful to learn a lot and practice a lot before choosing a boundary and deciding how to enforce it.  And Al-Anon advises us not to make any drastic changes in the first six months, so we can get the skills under our belt and be certain how we want to go forward.  So with the knowledge of the legal procedure, you might consider finding Al-Anon meetings (there are meetings online here if there are none in your area), reading the literature, reading all the threads on this board, and practicing the tools till you have a real sense of how and what you want to do with boundaries and your sons moving out.  It may be that you want to pay for their first months living somewhere else; it may be that you want to do it umpteen different ways.  Other folks in the meeting and here may well have ESH (Experience, Strength and Hope) on this issue that will suggest good ways forward too.  My own A was kind of grateful when I asked him to leave, as I think he was mighty tired of my bugging him about his drinking!  So who knows how it will play out, but with more tools you'll be equipped to take good care of yourself.  Hugs!

Thinking it through more, I'd guess that to set wise boundaries we need several things.  An idea of what a wise boundary looks like, from having seen or heard others' experience with boundaries.  The confidence that when we set one, it's right and we are right to stick by it.  (This confidence comes from seeing and hearing about others' experiences.)  And the tools not to lose our serenity when other people are trying to raise an almighty dysnfunctional fuss to get us to do what they want.  When we know we can hang on to our cool through anything, no one can control us any longer.  It takes a lot of practice and work on ourselves.  You're on your journey!

Give them a date, tell them your serious and if they are not out on that date, pack their stuff, change the locks and call the police if they give you hassle. They wont listen to your words but they will pay close attention to your actions. If you really want them out, its gojng to take some determined action.