Whether either of these sentences sounds correct, and if so, which of them, actually depends on your dialect. There's a good post about this on the website of the Yale Grammatical Diversity Project.
(Here's a sample map from them of the US -- they have a much better, interactive version in the article, but I thought I'd include this screenshot here just to have a visual. The redder a state is, the greater proportion of people there are who judged "I'm done my homework" as acceptable; the number represents the total amount of people who participated from the state.)
We can also consider a few more sentences:
"I'm finished" and "I'm done." Before today, I assumed these sentences would be generally accepted as grammatical by speakers of all dialects of English, but apparently "I'm done" has historically been more common in North America than in Britain. It does appear that these usages crossed the Atlantic at some point, and nowadays at least some British English speakers find them acceptable.
For many speakers (myself among them), "am finished" and "am done" can only be used intransitively, which means there is no direct object. For these speakers, sentences like "I am finished my sandwich" and "I am started my sandwich", with a direct object "my sandwich," sound incorrect.
But for some North American English speakers (apparently speakers of Canadian English in particular, but also a minority of speakers in the United States), "am done" and "am finished" can be used transitively. If the above sentences with "sandwich" sound correct to you and you're a native English speaker, you belong to this group.
For a subset of these speakers, "am started" can also be used (transitively, and I'd assume also intransitively).
Who says this?
The done my homework construction is a widespread characteristic of Canadian English, and it is also found in the United
States among speakers in Philadelphia, Vermont, and New Hampshire
(Yerastov 2008, 2010a, 2010b, 2012, submitted; Hinnell 2012; Fruehwald
and Myler 2013, 2014). It has not been found in the dialects of the
United Kingdom or elsewhere outside of North America.
In particular see
There is variation across dialects in terms of which verbs speakers allow in this construction. According to Yerastov
(e.g. 2010a, 2010b), there is a hierarchy along the lines of finished
> done > started. In other words, if speakers accept started (as in I'm started my homework), they will accept all three verbs. If
speakers accept done, they will also accept finished, but not
necessarily started. Finally, some speakers accept only finished. This
kind of hierarchy resembles the one found in the needs washed
construction with need > want > like (see here for further discussion
of the needs washed construction).