We are most likely "doomed" in that these changes are more or less inevitable. But we are not "doomed" in the sense that the language is actually breaking or somehow falling apart. Languages have been around for thousands of years, evolving and changing; no language has ever evolved itself into a corner or created a construction that makes the language non-functional.
Many of the "correct" English words and constructions that we use today got their start as stupid-sounding "mistakes". Our case and gender systems have almost vanished completely, for example.
In fact, if I am not mistaken, a couple hundred years ago, "the only people who passed the test were Anna and I" would have been the pedantically correct version, and "Anna and me" would have been the sloppy "wrong" version; it used to be that the copula would have nominative case on both sides (e.g. "it is I", not "it is me"). Maybe that puts things into perspective a bit.
If you find certain things silly, there is nothing wrong with feeling that way and avoiding them — and not all variations become mainstream. But, it is pointless to try to make any significant effort to stop these changes. Mainly because it is inevitable, but also because you'd only be protecting a momentary instance of a thing that is constantly in flux.
(Edit: I should also add that I don't even necessarily agree with you in the case of lay and lie. Most people don't even know the correct paradigm for conjugating these verbs. I interpret the changes in popular usage to mean that these verbs are undergoing regularization; the old paradigm seems to be inherently confusing and I welcome the change: the language is fixing itself. I thought this might also be an interesting point of view to consider.)
Addendum based on the question's first edit: I recommend reading up on Standard Arabic to consider attempts to stop language change in its tracks. Because Muslims believe that the Arabic used in the Quran is holy, they have attempted to maintain this version of Arabic. This Standard Arabic is the only "official" Arabic, it is the only one they learn in schools, and it is the only one they write in. But, in day-to-day use, you can't stop language change. Instead, every region has evolved a distinct dialect (and these "dialects" really stretch the definition of that word to its breaking point). In many cases, they are not mutually intelligible (for example, Moroccan Arabic and Baghdad Arabic), and nobody in the Middle East actually speaks Standard Arabic as a first language. Most people have an imperfect command of it anyway except for the most highly educated. Most of the time, if speakers of different dialects want to communicate, they speak in a simplified hybrid of their own dialects and Standard Arabic (leaving off things like case marking that exists in SA). The Standard form has become different enough from their spoken language that they can't actually manage to follow the rules automatically.
This also means that if I want to learn useful Arabic, I have to learn Standard Arabic to read and for some TV programs, and I have to learn another language in order to actually communicate with the people around me. There is not much practical benefit to this system (but they aren't doing it for practical reasons).
If for some reason we decide to freeze English as it is now and make a concerted effort to maintain this form as it is, we will inevitably end up with the same messy diglossia situation that they experience now in the Middle East.
Addendum after second edit: I can't keep up with this moving target :)
Best Answer
You are right to suppose that "a Sunday" is the most proper construction here. However, the most formal version of the sentence would then be
The use of "OK with you" already makes the sentence somewhat informal, so the lack of 'a' before Sunday is permissible and does not at all change the meaning:
Also, Sunday is rather being used as a generic name for a certain type of week day here (there are 7 'types' of which one is Sunday) as in
You don't need to say 'a pizza' here or 'a Sunday' because the 'a' is implied in the sentence. Some people would say "if Sundays are OK with you."
Regarding your concern about the meaning of the sentence "Which Sunday do you prefer, if Sunday is OK with you?", it would be typically interpreted as your asking someone whether Sunday is OK with them and if so, which Sunday would they prefer? If that is the meaning you intended then the sentence is very clear. If Sunday is not OK with the other person, they will let you know.