A vs ä ?
I know in different varieties of English certain words and sounds are produced differently, so giving words as examples doesn't always work so well. BUT you could try thinking about how you pronounce:
A - hard, half, far
Ä - can, that, land
Here are some examples that should work well for most English speakers (except ones with very heavy /æ/ raising):
A - father, spa, saga
Ä - happy, back, matter
Half isn't a good example, it's "haaf" in British RP but "hääf" in General American.
Hard, far are ok for a.
Ä = cat/rat/that are good.
That and o vs ö
And also vowel harmony: aö or äo don't occur together in the same word (except compound words), it's always a with o and ä with ö. (The full vowel harmony sets are aou vs äöy; i and e are "neutral")