Solved... with some uh, gentle... hints in comments.
I'm many a game piece, many a stain,
Oh... I'm actually a word quite full of pain.
Dice, Dyes, Dies
I'm many an ocean, I'm a good student's marks!
Oh... I'm actually quite small, and found in gardens and parks.
Seas, AAAs, Bees
I'm many a trashcan, I'm many a part.
Oh... I'm actually partial, and unfair at heart.
bins, bits, bias (thanks, Kate Gregory)
I'm a super popular drink, I'm a kid's cuddly friend!
Oh... you'll actually only use me after the end.
beer, bear, bier (thanks, Kate Gregory)
So, based on what those clues point to, the last line should be
I feel like I did something wrong with this riddle.
Additional comment by TheGreatEscaper:
In hindsight, the stanzas should be in a different order. The four words, BEES, BIER, BIAS and DIES are each exactly one letter off from the nonsense string BIES. Taking the letters that are 'wrong', and in order of position in the string, you get DEAR, as intended.