Yes, an ancova is recommended in your case.

it is a regression:

post = pre + group ( in that order)

Seen, 2006. Change from baseline and analysis of covariance revisited. Stistics in medicine. 25: 4334-4344.

Jos Feys The formula you give will test for mean differences between the groups after controlling for a single, overall rate of change. The next step should be to introduce an additional interaction effect to test for differences between groups in their rate of change:

post = pre + group + pre*group

I think you, your supervisor and other responders are in a sort of agreement about the form of analysis. It may be inappropriate for me to answer ignoring your experimental conditions (i.e. assumptions etc), the questions asked and the planned analyses. However, I guess your analysis (either ANCOVA or Repeated Measures ANOVA) comes under the general linear model ( perhaps, in your case, MANCOVA). Thus, it is a matter of specifying your response variables, factors or dummy variables and the covariates in a regression model. Interactions can also be tested in such a model.

I agree with David. One should test for equality of regression slopes.

see: https://discoveringstatistics.com/docs/ancova.pdf

For further info. The ANCOVA design with pre as covariate and post as outcome is preferred in a randomized design because this is a more powerful test of the hypothesis. The expected difference on the outcome (y) is the same for the model with and without pre as a covariate but the pre measure predicts y and this reduces the size of the error term producing a more sensitive (statistically powerful test). Adding other covariates known a prior to predict y should also increase sensitivity. The test of the effect of interest is treatment x post as noted by Jos Feys

Further points:

1. Centre your covariates (pre and maybe SES in this case). This helps interpretation as the coefficients will be estimated as the mean of any covariates (rather than an arbitrary zero).

2. If SES is categorical you might consider effect/deviation coding rather than dummy coding. If continuous, centre it. (Its arguably ordinal but that's trickier to handle in standard regression software). Centering is particularly important for the interaction model (or effect/deviation coding if SES is categorical).