"I get that /ʌ/ is used on stressed vowels and /ə/ on reduced vowels."
No. Yes, "/ʌ/ is used on stressed vowels", but there is no phoneme /ə/ in English. There is only an allophone [ə] of various unstressed vowel phonemes, including /ʌ/. (I am giving you my own idiosyncratic opinion here -- some linguists will disagree.) You're not going to be able to deal with the difference you're concerned about until you have some idea of the difference between phonemes and allophones.
Phonemes are perceptual, while allophones are actually pronounced. Unless they have some training in phonetics, or are just naturally very good at phonetics, English speakers don't hear schwa at all, because there is no such phoneme. They will hear a schwa as some similar vowel phoneme, probably caret /ʌ/. But there is nothing intrinsically impossible about hearing and pronouncing both sounds, it is just being stuck with the English phonological pattern that makes for the difficulty.
You have to get beyond the usual limitations of English speakers in order to control both schwa and caret. I suggest taking careful note of your tongue position as you say various unstressed vowels. I find that my tongue is retracted toward the back of my mouth when I say caret, but not retracted when I say schwa.