I'd guess it's a confusion because you wouldn't say "I feel my hair were straw" (you would say "I feel my hair was straw"), whereas in some languages you might use the subjunctive after a sense verb such as feel. In your examples you have an additional structure "as though" which commonly uses the irrealis/subjunctive, so it's different, but may be similar enough to confuse people.

An Ngram search for 'feel as though I were' finds plenty of examples.

It depends on whether you are using the word feel to refer to a physical sensation, or an emotional/intellectual perception where you might wish to express some doubt or uncertainty about it. You would say to a doctor "I feel as if my throat is on fire" but "It felt as though he were making fun of my hat."

What is governing the verb is the "as though" and not the verb feel. "He strode down the hall as though he were or was a soldier".

@Lambie: strode is not a verb of sense. And you could say to a doctor "I feel as though my throat 's on fire."

@TimR That's exactly my point. Please do not make a comment like the one you just made. How could I possibly think that stride is a sense verb?

@Lambie: What's exactly your point? You said "as though" governs the verb. It doesn't.

I feel as though my hair is straw. I feel as though my hair were or was straw. They all work.

grammaring.com/as-if-as-though

There's no subjunctive clause here. This is a simple case of irrealis "were" vs modal preterite "was". The difference is one of style level: "were" is here somewhat more formal than "was". "As though" is a compound preposition with a couple of content clause complements. The PPs are functioning respectively as manner adjunct and manner complement.

@BillJ I think there's more to it, at least in AmE, than formality or style. While a good percentage of speakers of AmE never use irrealis were, for those who do it can express a lack of commitment to an utterance, an uncertainty or tentativeness. "It felt as though he were making fun of my hat" is more tentative and uncertain than "It felt as though he was making fun of my hat."

@TimR I'd say that there is no semantic distinction to preserve. Here in the UK, irrealis tends to be favoured by older speakers (like me). Just as long as we don't call it the ill-named 'past subjunctive'!

@BillJ Well, perhaps you don't know what it feels like to be tentative and express less than 100% commitment to an utterance :-) Some languages mark for such situations and I don't see why English should be considered less expressive with its irrealis.

Irrealis and modal preterites both express modal remoteness.

@BillJ Although I believe you yourself always employ the term “modal preterite” consistently, there’s enough inconsistency between writers that you ᴍᴀʏ wish to clarify your own use of this term for our readers’ sakes. To wit I ʙᴇʟɪᴇᴠᴇ you always mean using the simple past tense w/o a modal aux. in a subordinate clause to show remoteness, a type of unreal modality, as in how I wish he ran faster represents a modality absent in He ran faster. Alas I’ve seen other writers who confusingly use the term “modal preterite” for such forms as did run, would have run, and even would’ve ran.

@tchrist 'Modal preterite' is the use of the preterite (simple past tense) where the meaning has to do with modality rather than past time: in He'd be upset if you knew the preterite suggests that you may not know, not that you knew. Is that clear enough?

@BillJ Yes, thank you. I was pretty sure that that was exactly what you always meant by it. There are just so many other strange-to-me uses of the same term out there that I figured it would help readers if stated explicitly.