"Old people do not hear well."
Translation:Starzy ludzie nie słyszą dobrze.
"stary" is masculine singular (an old man/human)
"starzy" is masculine personal plural. (old people)
'stare' and 'dobre' are either neuter singular or 'not masculine-personal plural'.
'dobrze' is an adverb.
'starzy' is masculine personal plural, and so would be 'dobrzy'.
The names of the plural actually say it all (almost): masculine personal is used for 'groups of people including at least one man', and not masculine-personal is used for everything else.
As 'people' definitely includes at least one man (otherwise it would be 'women'), it's masculine personal.
If they had good hearing, then [słyszą dobrze/dobrze słyszą] would both sound fine to me. But they don't hear well. And if you put "dobrze słyszą" together, then I guess the whole sentence was "Starzy ludzie nie dobrze słyszą". And first of all, negated adverbs are written together (niedobrze), but that would still feel wrong. You should negate the whole idea of 'hearing well' and not just 'well'.
If the sentence was positive (They hear well), it would be fine. But "nie dobrze słyszą" is wrong, because you should negate 'hearing well' and not just 'well'.
I put dobrze slysza, word order different but still acceptable. I remember the previous generation using this word order.