As harsh as it might sound but the masters back then had the rights on all their Dasis. The Dasis were allowed to have mate only with master and with whom they allowed(like wife can do that only with husband and those husband allowed) Although it was advised to avoid having kids out if such relation because it it happened then the kid as well as the mother would immidiately be freed of dasavta and would be raised as freemen. The master had the responsibility to nurture this kid in the same way as they did to his legitimate kids same food, same education everything but these kids didn't have the right to inherit.
Yes the master couldn't force himself over the Dasi but could definitely pester her, nor could he humiliate her in public. Probably this is the reason why Bharat ji stopped Shatrughan ji from punishing Manthara in public and instead imprisoned her.
If a master forced her in any way, or humiliated her in public, the Dasi would become free immidiately and would had the right to ask two things from her master as punishment (what Draupadi did by asking her freedom of her husbands and their weapons)
So Dhritrashtra here was doing exactly what was within his rights. He had consensual relation with his Dasi and when she begot a son, freed her and nurtured Yuyutsu the same way as he did his other brothers