It's the joint account and mortgage that causes your credit files to be linked, not marriage.
I'm in two minds about this. If you let him not sort himself out, he's likely to just get into an even bigger mess, where the interest bills each month will be crippling and any payments will not actually reduce the debt.
But if you pay it off for him, it's likely that you'll have to watch him like a hawk in the future, or he'll just end up overspending again. It's ingrained in a lot of people and very hard to change.
What is the car used for and who uses it? I would argue that if a car is mostly used to get to work and run errands, it is a household expense rather than a personal expense, but the car should be 'affordable' on the budget available. ie, if you can't afford an expensive car, you don't buy one.
Do you have your own car, or do you use other transport? Could he walk or cycle to work, or use public transport while you/he saves up for a new one?
In the past I've been anti-new cars on the grounds that they are very expensive and depreciate massively, but there are some very good PCP deals on city cars (eg Skoda Citigo) where the tax, insurance and fuel are very cheap and you don't have to worry about repairs or MOTs for three years. Even servicing is included. Would this be better than buying an old car that will use more fuel/cost more to tax etc?
Do you actually have 'enough' income to pay all your bills adn basic groceries etc and is it just a matter of him spending too much on luxuries?