Hi andwhatnow
Regardless of who stays in the former matrimonial home (FMH), Mamathulu is right - with a mortgage in your joint names you both have the legal obligation to your mortage lender to ensure it's paid, or both of your credit ratings will be adversely affected.
However, in practice it is a matter of whether your income less your outgoings/needs (including rent on your new home) leaves you with any surplus which might potentially be used towards the mortgage on FMH, and also whether your husband's income less his needs (inculding 100% of the monthly mortgage payment) means whether he has a budget deficit and so needs any financial help to pay the mortgage.
Another pratical financial factor which many separating couples need to take into account is whether FMH is significantly more (in size and cost) than is needed to accommodate the parent and child(ren) who remain living there. Although it is often preferable that FMH/child(ren)'s family home is not sold, sometimes it is a fact that the property is too big and so downsizing can help reduce mortgage payments and bills. I note that your FMH is already on the market to be sold anyway.
I recommend that both your and your husband try putting to one side "legal rights and responsiblities", and instead try focusing on the practical financial implications of you and your husband now living apart. From my experience as a family solicitor, this is more commonly how cases of this type are resolved, with the parties sitting roundtable themselves, and if needed also both parties sitting together with specially professionals (such as a mediator or collaborative family lawyers) in order to reach a balanced compromise agreement. I would hope the solicitor you saw flagged up all the options available to you to resolve your situation amicably, not just your "legal rights"? Go to the Resolution website to make sure you're getting the right advice from the right type of specialist family lawyer www.resolution.org.uk/
As for naming your child's "place of residence", this is not strictly required by law. I return again to the more appropriate focus for discussion being the practicalities of you and your husband living apart whilst your child shares time between you both. legal definitions of "residence" and "contact" are then not an issue.
However, for the purpose of the strict application of child maintenance rules, one of you would be the "non resident parent" (NRP) and so responsible to pay child maintenance to the "parent with care" (PWC). However, with proper balanced dialogue and an agreement (has happens with many cases), it's not necessary to involve the child support agency (CSA, but now called CMEC) at all but to have a voluntary agreement instead.
Further, who your child "lives with" can affect which parent receives child benefits and child tax credits. But, if there is proper balanced dialogue about how the distribution of all sources of income are used to pay the bills for 2 separate homes, then who actually receives benefits/tax credits becomes insignificant if the family as a whole (albeit separated) benefits from all sources of available money.
I hope this helps.