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I've seen the song listed as both "I'm Moving On" and as "I'm Movin' On", and I think it's more likely that the second is correct. It could be that it's meant to have the apostrophe and people just mistakenly say "Moving" instead, or perhaps the song has been listed (in some official capacity) as both at different times.
For example, here's a Wayback archive of a NYT article from 1999 that lists it as "I'm Movin' On", but I'm not sure what source(s) would satisfy the burden of proof given that this would impact the name of the article.