If the teen's account is 'in trust' does that mean that you or your creditors are not allowed to touch it? It might be worth asking Step Change a hypothetical question about this money. Obviously it is not acceptable to avoid paying debts by saying that money you have actually belongs to your DCs, but if there is a legal arrangement (the trust) this may be different.
You say you spent too much during a time of bad mental health, but what on? Do you have access to money? If you are not working due to being a SAHM, you should still have access to your partner's income (or other household income such as CB, child tax credit, maintenance from DS dad) in order to buy groceries, things for the house or DCs and your own personal spending money.
So yes, if the debts have been run up in paying for things that he has benefitted from, such as bills, groceries, household or DC expenses (is the younger one his?) then yes, his income will be taken into account when solving the debt problem.
But obviously if you have access to sufficient family money and you've overspent on your trinkets of choice, then it's different, but obviously your mental health is relevant as it's well known that this can cause overspending.
If you have no income and really don't want to involve your DP, one solution may be to offer your DCs money as full and final settlements - eg you can offer each creditor about half the money owed and ask for the rest to be written off (but start lower than this if the debts haven't been paid) and then you can aim to repay the money later on, if you can work when you are better/DC are older. But you really need to seek advice before making a big step in order to get the best solution for you.