This sounds like the situation my own parents are in, but my dad is an alcoholic (the "functional" kind with a good job, except there's no such thing as an alcoholic who is functioning, really).
@OP I think one of the key words you mentioned was drinking. Is it the eating that's more of a problem or the drink?
My dad used to be thin and athletic, but put on weight in his 30s due to a combo of things: stress, poor diet due to long hours at high-pressure jobs, drinking every day to decompress, quitting smoking (so replacing cigarettes with food/drink), sedentary job and no exercise, and a thyroid that quit (took years to diagnose and then get medication right).
My dad has been morbidly obese for years, and has a long list of health problems related to that. But it's the booze that holds him back from getting better. My dad is a gentle soul, but he is no longer at a point where he can manage 30 days without drink. Is your husband at that point?
If your husband is seeing a counsellor and taking meds for stress, that is a good step towards him taking care of himself. I wish my dad would take that step. That's a positive thing you can hold onto.
If your DH is on SSRIs, these can cause weight gain. Working through the roots of stress and core beliefs in counselling can make you feel worse before you feel better. You might see more emotional eating while he processes heavy emotions, before he gets to a point where he doesn't want food to be a crutch.
Alcohol can undermine the progress your DH might hope to make in counselling. It's a depressant and can interfere with deep sleep. Poor sleep can feed into anxiety cycles, which lead to more stress. So, if he can take a break from booze for the sake of his mental health, he might find that taking care of his physical health sort of falls into place.