Short and sweet. The longer the apology, the more room there is to add a 'but' clause.

Good: I'm sorry I left the door open so the dog could escape. I will make sure it's closed going forward.
Not so Good: I'm sorry I left the door open and the dog got out, but I was in a rush because I'm really stressed about X.
Bad: I'm sorry I left the door open, but if you had helped me with Y, I wouldn't have been in such a rush and this never would have happened.

If people don't like your apologies, try the simplest form you can and see what happens. If they still don't like them, it might be their issue, not yours.