Thanks for the fast response and suggestions. I will check out the projpred library.
To clarify, is this because of the location shift when standardizing, i.e., subtracting by the mean? Is there a way to preserve/extract the information about the effect direction?
I wondered if there is some more advanced method, for instance, using posterior predictions, that can then also tell whether A and B have a positive or negative effect. Standardizing the data by means of subtracting the mean and dividing by the standard deviation seemed to me like a perhaps too simple way, given that A and B might not be normally distributed. It also seems more intuitive to me to interpret effects on the response rather than the predictor scale.
Btw., I saw a related topic, and it seems that variance explained might not be a better alternative.