"I changed the bedsheet."

Translation: Zmieniłem prześcieradło.

[JamesTWils]

So, are these simply two variants of the first person singular ending? Are they regional variants?

[immery]

They are gender variants.
Zmieniłam is female speaker. (an usually -łam ending)
Zmieniłem is male speaker. (and usually -łem ending)

we have this distinction also with you singular, we ,you plural.
on/ona/ono have masculine/feminine/neuter variants
oni/one have their own respective variants

take a look on the chart to see what I mean
https://en.wiktionary.org/wiki/zmienić

[JamesTWils]

Now I see the similarity to other Slavic languages. I have just never seen the gender ending and then a personal ending after it. The lack of tips and notes is making me ask far more questions in discussion than I usually would.

[immery]

I saw someone comment that it is because the last part is actually moveable,

not an experiment for beginners, but sometimes -m/am/em; -ś/aś/eś ends up at the end of something else than verb.

[mihxal]

Np. "Coście skurwysyny uczynili z tą krainą?" ;)

[JamesTWils]

Well, considering Russian, Czech, Croatian, and other Slavic languages, it looks like the -l(e)-, -la-, -ly- is a sort of adjectival ending, originally on a past participle. Polish then adds the conjugational ending of verbs after that.

[GennadyTel]

Przed chwilą było powiedziano, że ona nie mogła pić. Czyli więcej nie mogła? ;)