I can add that the variance was more than twice the mean. I looked into over-dispersed Poisson models, but traditionally my field leans heavily on normal models. Is there a way to decide on the type of model?

Edit to add more information based on comments:

I had two groups of participants find a target. I measured distance travelled (the program I used has an arbitrary measure of distance). I measured each participant over the course of three trials. I predicted a group by trial interaction (I expected one group to improve more than the other groups).

I anticipated using a multilevel model using LMER in R:

    model <- lmer(distance ~ covariate + trial*group + (1 | participant), 
                        data = df)

Here is the distribution of all scores:

and the distribution by group: I guess my major concern was that I wasn't sure if it was appropriate to assume a normal distribution. The residual plot showed did not seem 'random' to me, which is what caused my hesitation.

I guess while I am this far in, my secondary question is, Can I include the effects of trial as a random slope without including the interaction as a random effect, such as:

    model <- lmer(distance ~ covariate + trial*group + (1 + trial | participant), 
                        data = df)

Thank you!