Looking for etymology or information on the alternate meaning of "I don't care to X"

I think it's just a corruption of "I don't care about getting dirty", similar to how "I could care less" is often said when the user obviously means "I couldn't care less."

So, we have (yet another) expression (such as “sanction”) with antonymous meanings. The antonymous meanings need not occur with equal probability. When the probability of one of the antonymous meanings is sufficiently low (from the reporter’s point of view – it might be quite high within a given locale), it is prone to be regarded as an anomaly when encountered. The case in point is analogous to the fact that “careless” can have the two (near-antonymous) meanings of “sloppy” and “without a worry”, the latter being a nearly archaic meaning.

Which of the two antonymous meanings is operative depends on what “to” binds to. The expression “care_to [raw infinitive]” is a softer way of saying “want [full_infinitive]” (for example: “Do you care to go to the carnival with me today?”), whereas “care [full_infinitive]” means “have a worry about [corresponding gerund]” (for example: “Do you care to get dirty?” means “Do you care about getting dirty?”).

Consider the following two passages:

“Get dirty? – I don’t care.”

“Get dirty? – I don’t care to.”

The first is unambiguous, but the second is ambiguous, depending on what “to” binds to. If folks in a region do not bind “to” to “care”, it can be confusing for visitors from the wider world where “to” DOES get bound to “care”.

Having grown up in an area where the positive sense of "I don't care to VERB" is common the best equivalent of this in standard English is more at "I don't mind VERBing" rather than "I don't care about VERBing". I have no sense of the etymological history of the phrase, and I suppose it makes sense that it might have come from the 'care about' pattern, but there is a subtle difference in meaning:

I don't care to pet-sit your cat while you are away.

which means the person is positively agreeing to do so even though some people might consider it too onerous a task to undertake. That doesn't have the same meaning as:

*I don't care about pet-sitting your cat while you are away.

I'm having trouble giving a good interpretation but it seems to have either a neutral or mildly negative sense.

I remember encountering this usage in Southern Indiana hill country, where there is a considerable influence of Kentuckian words and phrases. When you say it's a "southern" usage, where specifically have you heard it?

Another phrase I remember from there is "Gimme some o' them cheese, Hoss", which meant "I'd like some cheese, Horace", said to a storekeeper. "Them cheese" was also referred to as "them rat cheese", an orange Colby-type cheese.

I'm from the Southeastern United States. I think the usage of I don't care to to mean I don't mind comes from it being the response to a question. Ex: Person 1: Would you care to move you truck so I can get out? Person 2: I don't care to. In this exchange person 1 is making a request which person 2 then agrees to. By "I don't care to move my truck" they mean no, I don't care which equals I don't mind Once this usage pattern is established it becomes easy for the phrase to take on an almost opposite meaning to the original negative meaning. Especially when it comes to responding to a request, which is usually how I hear I don't care to used to mean I don't mind rather than I dont want to/ don't like to.