in sickness and in health

There's a whole study in a medical journal trying to figure out what disease Tiny Tim had in A Christmas Carol that

crippled him, but only in one leg;
would kill him over the course of a single year;
could be healed with 19th century medicine if a sufferer had access to enough money.

The smart money is apparently on either rickets or distal renal tubular acidosis, the treatment for both basically being oranges and beach vacations (two things a rich guy could provide).

iirc Glenn Close had to die of poisoning in "Hamlet." She did her research and decided that she was being killed by strychnine
Can I get a prescription for beach vacations from my doctor?
- As far as I know your doctor can prescribe literally anything. Whether or not you can get your insurance company to pay for it is another question

It's probably tuberculosis. Seaside trips were a classic part of the doctor's recommendation for consumption.

There's even some evidence that vitamin D (which is produced when your body is exposed to sunlight) may help slow TB infection.

https://www.harpersbazaar.com/uk/culture/g24261075/actresses-wearing-corsets-period-dramas/

The clothing alone was enough to kill a weak woman!!

Not really. Hollywood has propagated a lot of myths about corsets, partly because of incorrect application of them on Hollywood sets.
- Call me crazy, but I feel morally justified in not watching a 15 minute video about women's underwear. But I will accept the idea that modern folks might not have understood the right way to wear corsets.

I have no idea what "classic literature" this refers to?

If you want to approach it from the opposite angle, check out John Green's Everything is Tuberculosis.
In the Pre-Industrial Revolution period, TB was extremely commonplace and seen as a romantic disease that killed slowly but made the sufferer more beautiful and artistic. Different theories said that going to more humid (like seasides) or dry environments could help one treat or beat the disease. In reality, this sometimes worked because TB tends to be more common in population centers and the real aid was just getting away from others.
- Reminds me of the magic mountain. Its a big read but a good one
I haven’t read much but this is definitely a thing in Jane Austen’s works.

Were houses by the seaside less packed with toxic substances?

If we are talking about post-industrial revolution times, the air of cities was incredibly polluted, so getting out to the sea with its strong winds would definitely provide solace. I wouldn’t know about houses, hopefully someone can come to enlighten us

Literaly in pokemon ruby, Wally is ill and moving to a town with fresh air cures him.

https://www.youtube.com/watch?v=THc1X-kEX4E