I would not do this. They have insanely high expenses, 40K in debt in addition? They are not going to get things under control through anyone gifting them large amounts of money. And asking not just one, but TWO relatives for the total amount of $60K? That is insane and they are pretty terrible (and yes, likely desperate) for thinking this is anything anyone should do in these circumstances.
The idea that it could be awkward if you ask to do alternatives (like paying the debts directly) instead of just giving them money? That's a feeling to pay attention to. You're uncomfortable with calling them out for being bad with money, but they want YOUR money because they're bad with money. You NEED to listen and express that feeling. They may have things in crisis mode due to the disabled monkeywrench that has now thrown off their delicate balance of spending more than they earned because they can't pretend any more - but it's still a problem they made before they became disabled. They just have a great reason now for why they need help and feel safe in asking others to bail them out, so they're likely going to use this as a "get out of jail free" card and hit you and the other relative up and think they'll dump their problems on the responsible ones they know...
SO if I cared about this relative, I would offer to help them, but I would not give them anything near this amount and I would not give any money directly to them.
I would tell them as a condition of the "loan" you need to have them lay all the cards on the table: exactly what they spend, what they owe to EVERYONE, if there is stuff they can sell, help them sell it, cancel all extras (cable/subscriptions/eating out/fancy store). They need to make serious cost cutting efforts. They're in hair on fire debt (which did NOT happen since July), spend too much and just became disabled? If they are doing the work, then I'd be more inclined to help them dig their way out, so are they?
I would possibly help pay down the credit card debt (take over the monthly payment myself so it IS paid, but not gift any money directly to them) and they better damned well not be adding more to it. I would also expect them to cut expenses to only that which they need to survive: mortgage/rent, insurance (health, car & home), utilities, basic food budget, cheap phone service, gas money to get to store/doctors... no eating out, no buying crap, no subscriptions, no nothing. They don't have the money to spend any more. Until they get their shit handled, why should you have to clean up their mess?
They're disabled now... continuing to act as if they are capable of dealing with this now when they didn't when they weren't disabled is delusional. Don't help them keep being this way because it is not a kindness. Unless you want to spend the rest of your life (and lots of your money) supporting them, you have to have a serious talk about how much you will help them, but they have to do the real hard work themselves.