A mixed model, also called a multilevel model or a hierarchical linear model does sound like a good choice (though with only 2 observations per participant, a mixed ANCOVA would probably work just as well, or a single-level model with clustered standard errors).
However, if you use multilevel models it would probably be good for you to first read a bit about them in general if you are unfamiliar with them, but how to do it: in SPSS you first need to arrange your data in long format so that each participant has 2 rows (if you have 2 observations per participant), like this
id scenario dv trait
1 high 3 3.33
1 low 2 3.33
2 high 5 4.11
2 low 1 4.11
...
Then, you can use SPSS's Analyze...Mixed Models...Linear and first put your participant id variable into the "subjects" box. You can likely ignore the repeated box (it's for specifying a residual correlation structure for different time points, mostly relevant for longitudinal designs). Then just use the model specification menus as usual. Remember to tick the "Statistics...Parameter estimates for the fixed effects" that gives you the fixed regression coefficients for the scenario, trait, and their interaction.
The following syntax should also work:
MIXED dv BY scenario sex WITH trait age
/FIXED=scenario trait sex age scenario*trait | SSTYPE(3)
/METHOD=REML
/PRINT=SOLUTION
/RANDOM=INTERCEPT | SUBJECT(id) COVTYPE(CS).
COVTYPE(CS) means that the within-person correlation structure is compound symmetry, meaning that within-person correlations between all time points are assumed to be equal (for a given person); that is enough here as you only have 2 time points per participant.
(In longitudinal designs you may want to specify within-participant correlation structures that take into account the temporal closeness/non-closeness of observations, but that's not relevant here.)
EDIT. So, I don't love multilevel models for this type of designs with 2 observations per cluster (here: participant), they are a bit of an overkill. The simplest thing model-wise would be to use clustered standard errors, but I'm not the best to advice on them because I never use them (they are "not a thing" in my field) and I haven't even tried using them in SPSS (only in R). And unfortunately, it seems that SPSS tutorials for this are not the best. I think "SPSS Complex Samples" is a relevant search term. If you can use R, there are plenty of pretty clear tutorials.
However, if you have any experience in conducting ANOVAs, better yet repeated-measures ANOVAs in SPSS, I'd consider using repeated-measures ANOVA with the personality trait as a covariate. It's simple and gives you what you want and is basically designed for situations like yours. Unlike with multilevel models you need to have your data in short format , i.e.
id high low trait
1 3 2 3.33
2 5 1 4.11
...
and then use Analyze...General Linear model...Repeated measures...
and put in your repeated factor as having 2 levels, click "Define", put the "high" and "low" variables into the "Within-Subjects Variables" box and the trait into the "Covariates" box. The analysis enters the interaction between trait and the high-low variable as a default, and takes care of the non-independence of your observations. See this tutorial, though they have an additional between-subject factor in the example ("Exam") that you don't have, ignore that.
However, note that repeated-measures ANOVA/ANCOVA assumes sphericity - equal variances across all conditions. Be sure to check that this applies if you choose this test. See here for explanation of sphericity.
