Of the two, I'd suggest divorce proceedings. There srr so many stages to it, and you can stop it at any point, so you might as well stsrt, so, if he doesn't improve (likely) you have already made progress in the right direction.
BTW, I'm assuming UK, hut if you're based elsewhere, some of this will need to be honed.
You've taken some legal advice, so this might not surprise you, but as you're married, its probably immaterial that he hasn't contributed much/anything to the house; it's a marital asset and will go into the pot. Can you afford to buy him out? Talk to your mortgage people, maybe shop around for a re-mortgage.
The main worry you might have is, as you work full time and he doesn't, would he be able to establish that he is DS' primary carer? So you need to start building your evidence: who does school runs, cooking, cleaning, DS' clothing shopping, bath and bedtimes, children's parties and activities, who provides most/all of DS' emotional support, etc. See, it's not going to be enough to just say "I'm the mum and he's an alcoholic", sadly. You need evidence.
Any evidence of domestic abuse/violence needs to be noted as well. General bad behaviour, its effect on you and DS (you've done a very good start in your OP). Has his alcoholism put DS in any immediate physical danger, for instance? Has police ir docisl services ever been involved? Etc. If you can build up a timeline it'll help your case, not to mention your own clarity!
Talk to your GP wnout how you are feeling; start to get a paper trail going (and GP might be able yo help you access services, counselling, etc). Talk to Women's Aid or your local domestic abuse people (check with your local council); they can hellp with all sorts of emotional and practical support, and will probably have access to specialist divorce solicitors. Esp if there's any domestic abuse involved, you do not want a genetic solicitor - you need one who understands and even anticipates what your DH might be capable of. If you think there's anything to report to your local police domestic abuse team, do so.
Collect up paperwork: bank accounts, mortgage, savings&debts, pensions. Marriage cert, birth certs, passports, etc.
As to who moves out, well, you make a good case for you staying, but what could be your H's case? See above re: primary carer. You might have to both stay in the marital home until you can convince him to move out. Him having any other place to stay isn't relevant (or your problem), but it might, yes, be helpful to give him enough money to set himself up elsewhere (obviously making a note of this in your eventual financial separation).
However, if you fear violence, do take steps to protect you and DS.
Hope that's enough to get you started! 
Best of luck.