Oh my goodness. I'm so sorry you're going through this. And in your defense I will say this.
This situation is inappropriate at best and sick and twisted at it's worst. I do not think any woman would have this cavalier attitude of the 19 year-old being an adult and it's all good. Um, no. This is not a 19 year-old woman this man has met on this street. Yes, that's happens all the time, but this is NOT the case. This a 19 year-old woman that was introduced to the family at 16 years-old as the babysitter, then later on as a pseudo-sibling. There is a HUGE difference in that. I am 35. There is a 16 year-old boy that walks his little sister to the bus stop every morning. I could never imagine myself thinking, 'Hey he's cute. Let me hang around and woe him until he's of age." OMG, yuck!! I really, really wonder what woman, or man for that matter, would have this devil-may-care attitude when it comes to their own family. Good grief.
Secondly, I do not think you need to start pointing fingers just to justify you having sleepover's of your own. I think if your ex were having sleep overs with a woman that wasn't the old freaking babysitter who was 16 at the time of introduction, and carrying on a relationship that is weird and inappropriate in front of your children, I would say you're probably looking for a reason. But that's not the case! Also, I don't think you would post this question because it's of the same caliber of what you're doing. Is it right? No. But that other mess is just wrong!
With all that being said, what can you really do? If anything, you might as well drag the issue all the way out the closet and let the kids, the family and whomever else is involved know the truth. At least it won't be so sneaky, and perhaps it will blow off some of the weirdness. Unless they're getting married, no co-parenting. Would you do that with anyone else at this stage of their relationship?
The next thing I would suggest is control your own behaviors. I know your moving toward marriage and you're in a serious relationship. But maybe you shouldn't have the sleep overs because so much is going on already. You and your significant other are grown adults as well as parents and can stave off any further confusion for your kids and be able to deal with that on a mature level. You can blend families without sleepovers. I'm sure he'll understand. I think there is too much going on for your little guys to take in right now.
So in short, stop your own sleep overs, explain the relationship, hope for the best, and brace for the worse. I really hope everything works out for you.