Translation:I am not moving over, but changing clothes.
If you're trying to make sure the difference between similar sounding words is heard, like here, in the languages I know you would stress the syllable in which the difference lies: átöltözöm. The lady doesn't do this. Wouldn't you do it in Hungarian?
I don't think "moving over" is the right phrase here. What is? "Changing houses"? :-) (Likely not.)