Pedro,
Instead of saying that you want to include the covariate in the analysis, it's better to consider why you want to include the covariate and how it changes the interpretation. You don't necessarily want to regress out covariates, especially when the covariate is different between groups.
DO -- Different offsets, no covariate/no slope --> the offsets are the group means not adjusted for covariates.
DOSS -- Different offset, same slope model --> If you don't subtract the mean, then the offsets are the group means when the covariate is 0 for all subjects. If you subtract the mean, then the offsets are the group means when the the covariate is the mean covariate for all subjects. The difference in offsets won't change with mean centering the covariate.
DODS -- Different offset, different slope model --> Not possible because the covariate for controls is collinear with the group term.
There are other models that you could construct - such as only including a covariate for the drug group, but the interpretation will once again be different.
The bottom line is to decide what you want to test and how you want to interpret the results before adding covariates.
With DOSS, you will reduce/increase the group differences (depending on the slope of the covariate) because you are interpreting the results when the covariate is 0 in the drug group.
Best,
Donald