I heard that adjusted R-squared from two model are comparable only if two models use the same response variables. So if the response variable in one model is Y and the other one is log(Y), then I should not use adjusted R-squared; rather I should use AIC, BIC, etc instead. Is this true?
I believe using R2 or adjusted R2 is okay in your case. Fact is that we should not use RSE(Residue Standard Error) when the scale is different. This is because both R2 and adjusted R2 are normalized quantity having maximum value of 1, but the RSE is not normalized.