No, it is not okay for the priest to “make exceptions”. “Making exceptions” does not fall under the binding and loosing power or any other power a priest might have.
Regarding the priest telling someone it is all right to miss Mass on Sunday if they absolutely can’t get there (for example due to their work schedule), the person would have to ask the priest in advance of missing the Mass and the priest would give them a dispensation from their Sunday obligation. A dispensation is a relaxation of a purely ecclesiastical law. It’s not an exception allowing someone to commit sin, because no sin is committed if the person gets the dispensation properly. The priest is allowed to grant certain dispensations and the bishop is allowed to grant others. (There are also situations where the Church teaches that you don’t have to speak to the priest in advance in order to miss Mass, such as when you are too sick to attend Mass. In that case you are automatically excused.)
A priest giving advice on things like birth control or living with someone after divorce does NOT fall in the same category as dispensations, because those things don’t involve purely ecclesiastical laws. The priest is not allowed to give somebody a “dispensation” to use birth control.
When a priest gives advice on birth control or divorce that seems to run contrary to the teachings of the Church, it’s usually because that priest has decided himself that he doesn’t think using birth control, living (as in having sexual relations) with a new partner after a divorce, or whatever is a sin, and therefore he gives other people advice based on his own opinion, not on Church teaching. I personally have not experienced this situation, but I know others who have. This is why it’s important to be educated about what the Church teaches, so you can know when your priest is telling you something that seems really wrong. If you suspect you got wrong advice from a priest, then you should ask him for a full explanation, and maybe consider talking to a different priest about the same issue.