Ok that's fab news!
If you're on a rolling tenancy, all you need to do is give one complete month notice. This means one complete pay month. So if your tenancy started on 2nd, then give notice you give before the 2nd and it will end on the following on 1st of next month. If you're giving notice after your start date, then it'll still run to the start date of the 2nd rolling period to fit in one complete month (unless you have a nice LL who would allow you to do something else). I hope that makes sense
You need to give notice in writing to your LL if they are acting as their own agent. You do not need your DH's agreement to end the tenancy early if it is in both your names. My advice though would to be to call the LL in person first to give the heads up that the tenancy will be ended (1 complete month isn't much notice despite being the lawful minimum) because in doing so you can frame your phone call as a heads up you're both living together but are now separated so the housing situation will be changing soon and that you wanted to ensure they get as much notice as possible before the official notice landed on their doorstep (which puts you in a favourable light) then talk through your options at the same time (although LLs acting for themselves may not know without research!)
So now you have some thinking to do before you give notice to end the tenancy as you don't want to inadvertently end up homeless because you've run out of time to sort out your next step! What do you want to do from here? Did you want to stay and keep the tenancy in your sole name, or leave and find a new home?
Asking your LL for a new tenancy in your sole name has advantages and disadvantages. You'd need to explain to your LL your financial situation and assure that you can meet the costs alone. The less people there are in the tenancy, the greater the financial risk for the LL. You can gently point out that this option would benefit your LL too as there'll be no costs to him through the property being empty, nor cost of prep of the property for new tenants (eg clean and decoration). Bear that in mind if he needs a little push to accept you! You'll get a new fixed term tenancy in your own name before it reverts to the rolling as it would be a new tenancy, not a continuation of the last, so you're tied in until the end of that period.
If you are able to be sole tenant on a new agreement the biggest disadvantage is that your DH may not want to leave! Your LL should confirm in writing to your DH that the tenancy is ending and vacant possession is required. He could be an arse though so just a heads up.
On the plus side, you save on moving costs and it's less disruptive for you. There will still be a need to shift to a new tenancy agreement, new deposit in your sole name, new inventory, LL may want to do a reference and credit check to assess your sole ability to pay (many won't but you can ask what he'd want to do).
If you don't want to stay, it takes time to find a new place so allow for that with your notice period. You could ask the LL to finish your tenancy mid month if that fits in better with the start date of a new place. Don't forget to allow for fees and a deposit for your new place, as the release of your current deposit will happen AFTER you've already moved.
Just as reassurance, although you only need to give one complete month to end a rolling tenancy, your LL needs to give 2 complete months before a section 24 is served, notice that the LL wants his property back, and that is just a a request to leave and not legal possession notice so you'd have more time after that even (I can advise more if it got to that).
I hope I've made sense and it helps. I'll keep your thread on my watched list in case there's anything I might be able to help with.
Time to start thinking and get planing. I hope you've got a lovely LL like me! Best of luck