My given name is Anthony, and I have this problem. I grew up in Australia, and visit some family in Poland every other year. The Australian pronunciation of my name is particularly incompatible with Polish, so I change it from 'ænθəni' to (something like) 'ɑːntɒni'. The 'æ' sound at the start of my name is totally foreign, and if I start introducing myself in Polish, then say my name the way an Australian would, the entire sentence just sounds too weird. Obviously Polish people are familiar with Western culture, and have probably seen the name before, but it just sounds too strange when used in spoken Polish.
The sound [æ] is not entirely foreign to Polish and some other Slavic languages. It often shows up as an allophone of /a/ when it's between two "soft" (palatalized) consonants (e.g. the first /a/ in the word "niania"). The problem for us learning it is that it's not a separate phoneme, and worse yet, the environment in which it occurs in Slavic languages doesn't correspond to anything in English - and, conversely, in English it appears in environments where it could only be [a] or [ɛ] in Slavic languages.
The name itself is, of course, originally Roman, and it's also the name of many Christian saints, so basically every Christian country (not even necessarily a Western one) will be aware of it and have some version of it; for Polish that's be "Anton", I think, same as in Russian.
This doesn't happen in my speech, I certainly pronounce both <a>'s as [a] in "niania". [ɛ] is different, it certainly becomes [e] after palatalized consonants. I agree that people cannot tell the difference intuitively, though.
It happens more in fast speech. If you draw out the vowel - as people tend to when they are trying to get a better feel of it - it will end up at [a] even if it doesn't start there. I suspect you'd need to actually record it and then look at the formants to tell for sure.
