Which test is appropriate?

I'm trying to test whether time (IV1) has an effect on the relationship between a specific personality feature (IV2, quasi-continuous) and scores on a social cognition task (DV, quasi-continuous). Participants are placed into either a timed or an untimed condition of the social cognition task and the personality feature is analyzed using a self-report measure.

I was originally wanting to do a regression but I do not believe my data meet the assumptions (although to be honest I've only done a lack of fit test). I'm not very experienced with statistical analyses so any help would be greatly appreciated!