I work for Citizens Advice.
What kind of tenancy agreement do you have, is it an AST (Assured Shorthold Tenancy) and is it fixed term or periodic? Check the tenancy agreement to see if there are any clauses about changing from joint to single tenancy, and about claiming benefits (HB/UC are obviously the relevant ones).
Some tenancy agreements specify that tenants can't claim benefits and that's usually because the landlord's mortgage or insurance forbids it. Your landlord probably has insurance even if they don't have a mortgage.
I don't think you necessarily need to tell your landlord that your husband has moved out, or get his name taken off the tenancy agreement (unless it states that you should). So if you do have to tell your landlord you are claiming UC, you could just say that it's due to a change in circumstances, without specifying what those circumstances are. You certainly don't have to tell the landlord which element of UC you are claiming (personal, child, housing allowance).
You should claim everything you are entitled to btw and as PPs have said you can't pick and choose anyway.
My advice is to apply for social housing - there'll be a wait so it's wise to get on the system ASAP - and also ask your local council about private landlords that accept tenants with benefits. I'm not saying you will have to move (or that you should) but it's wise to consider other options in case it comes to that.
It is important that you can afford your rent and essential living expenses.
For more advice I suggest you visit or contact Citizens Advice. Take your tenancy agreement with you or have it in front of you when you call.