Rather than two separate cases, consider if it were a "something thing", rather than a "things thing" versus a "thing thing".
The primary function of the compound term is to convey the initial "thing". For example, there is no doubt what those things generally are in either of the two provided examples ("admissions office" and "ladies room"): (1) an office and (2) a room. The modifying noun should further clarify the purpose of the "thing". If the modifying noun instead creates doubt or confusion as to the purpose of the "thing", that particular combination is likely avoided due to a failure to properly communicate the specific "thing".
Consider if the initial construction is interchangeable with this construction: a "thing for something". Rearranged: (1) an office for admissions and (2) a room for ladies. Erroneously, with modifiers made singular: (1) an office for admission and (2) a room for lady. The purpose conveyed by the singular modifiers fails to properly explain the purpose of the "thing" to the audience due to ambiguity. Given either construction, is it an office for admitting to something specifically (i.e. self-incrimination)? Given the first construction, is the room (nonsensically) a lady, or maybe is the room made by someone called "Lady"? Given the second construction, is the room for someone or something called "Lady"?
What about singular modifier examples, like "word salad" or "absentee ballot"? The second construction flips plurality: "a thing for [many elements of] something". Rearranged: (1) a salad for words and (2) a [type of] ballot for absentees.
Consider a final construction that recognizes the clarified purpose of the "thing" is specific (so that the purpose applies to a specific set of people, places, and things): "for a singular something, this thing". It's important to note that this construction demands only a part of the whole, forbidding the usage of the original "something" as a whole. Rearranged: (1) for an [application of] admission, this office, (2) for a lady, this room, (3) for a word, this salad, and (4) for an absentee, this [type of] ballot.
This last construction is resistant to the erroneous singular form usages because of the refusal to allow the "something" to remain whole. The previous, erroneous examples ("an admission office" and "a lady room") rearranged: (1) for a part of an admission, this office and (2) for a part of a lady, this room. While both statements are technically true statements, they mean something completely different than the intention, which is a failure to communicate (and should be avoided for the sake of clarity).