I performed two studies on the relation between the stimulus valence (positive/negative) and the action tendencies (approach/avoid) measured with TMS motor evoked potentials technique. For each study I performed a Repeated Measures ANOVA with the 2 within factor (the stimulus valence and the action tendencies). I got a highly significant interaction in the first study and just a tendency in the expected direction in the second study (p=.08). Thus, we decided to do a mini metanalysis to present the results in a scientific paper. Both the studies have the same design, the same experimental question, study 2 is different from study 1 just for the manipulation of a variable concerning the experimental setting and also the samples are independent. I would like to know whether it is correct to calculate Cohen d as effect size in each study, the correct formula. Moreover, once that I have the effect size for each study, I guess I have to average the two effect size, do I need to weight the variance in the mean effect size?
You should have all the information you need already as you should have the coefficient for the interaction and its standard error. If your software has not provided this you may need to refit your model using a mixed effects model with a random effect for participant. Then with your two coefficients (which are your effect sizes) and their standard errors you just plug them into standard meta-analysis software and obtain the summary coefficient with its standard error. My only reservation about this is whether your design is balanced. If it is not then you need to extract the coefficients for the main effects as well and their variance-covariance matrix and use multivariate meta-analysis.