If the sentence isn't expressing a lack of feeling in my nose, then what does it mean? I can't see a conceptual difference between feeling and being able to feel. If someone asks you, 'Can you feel your legs?', it does not make any sense to reply, 'Yes, I can feel them but I don't feel them.' The difference, it seems to me, is simply idiomatic: in English we don't say 'I don't feel my legs', but 'I can't feel my legs', but the meaning is the same.
Mothaigh can be used for both the “touch” meaning and the “sense” meaning (among others, as can the verb “feel”), but Irish doesn’t express “I cannot X” with Ní X mé. If you’re only referring to the English idiomatic usage, it would depend upon the particular meaning of “feel” being used in the English question; a reply of “Yes, I can feel (touch) them but I don’t feel (sense) them” could make sense, but might require some sort of intonation to get the point across, if the responder wasn’t sure of which meaning the questioner intended. “I don’t feel my legs” might not be used as often as “I can’t feel my legs” — perhaps my legs have severe scarring from a past near-fatal accident that I don’t like to think about — but it can certainly be said in English, so “don’t feel” and “can’t feel” can have different meanings. (This is more obvious for the “touch” meaning, but it could also be said for the “sense” meaning, e.g. if I purposely distract myself to ignore whatever might be happening to my legs, “I don’t feel my legs” could still be distinct from “I can’t feel my legs”.)
I realise that the English word 'feel' has the 'touch' meaning as well as the 'sense' meaning. I just didn't think that 'mothaigh' carried that meaning. Ó Dónaill has 'Mhothaigh mé faoi mo chosa é' - 'I felt it under my feet', but this describes a sensation in the feet caused by the stuff with which they are in contact; it is not really the same sense as 'Someone felt my bottom while I was waiting at the bar', for example.
So I take your point as regards the English word 'feel', but I don't think that this example can mean 'I feel (i.e. touch or rub) my nose'. The examples in the dictionary all seem to have the sense of being aware of something, rather than the sense of touching. Can you show me any examples of 'mothaigh' being used in the latter sense?
Also, I'm not sure that Irish doesn't use 'ní x mé' to mean 'I can't x'. What about 'ní thuigim'? If you don't understand something, doesn't that imply that you cannot understand it (right here and now, not as a potential result of a course of study, say); and if you can, in what way do you not? To me, it seems the same as 'feel' (in the 'mothaigh' sense of sensation or awareness, not the wider application of the English word) - these are autonomic actions, not deliberate: if someone kicks you in the shins, you don't think about whether or not to feel it.
The NEID offers six translations for “feel” in the sense of “touch”, with mothaigh listed first. The first explanations of definition 1 a of feel² in de Bhaldraithe are Mothaím (rud) sa láimh; láimhsím. Definition 1. (b) in Ó Dónaill has two such “touch” examples. Dinneen defined mothuighim (the older spelling of mothaím) as
I feel, perceive; know; touch; I feel the loss of, miss; I hear
Regarding ní thuigim, if I had never seen a telephone before, and you asked me if I understood how to use one, I’d say that I didn’t understand; but if I immediately saw you receive a call and speak to someone using one, I couldn’t honestly claim that I could not understand.