Okay thank you for the additional information. It's great that the kids get along well. They are also living in separate households as well, and not being parented by Pete etc at this stage. The hovering you describe is clinginess and sounds over the top, which is why the kids sense it's uncomfortable. It probably gets old for you as well to have him all over you like that.
It sounds like Pete has insecurity issues which are his own to work through. The situations you describe are irrational such as needing to be at a business meeting where there are men, etc. It is not simply that he feels other men are on the make, it sounds like he also feels that you might leave him for someone else, and as there is no evidence for any of this, it really goes back to Pete needing to work through this. The fact that he gets jealous when you put the kids first, is a major red flag! You will be needing to do this quite often as life happens and kids need you, and kids come first!
I think you are wise to not marry until you are completely comfortable with these issues and feel they are resolved. It's great that you are starting counseling and I feel it's necessary and will be very beneficial for you.
The deadline place for engagement will more than likely be dropped once counseling begins, as April 4th is soon approaching and you will need time to see if these issues can improve. If Pete insists on the deadline, I would most certainly NOT allow that to influence you and simply let him go, if he sticks to the deadline. Counseling is what is needed here, time to then see if things can improve. If they do, perhaps you will decide to marry. If they don't improve or not enough to where you feel really really comfortable, and Pete doesn't want to just keep dating, then you may need to move on from the relationship. Please click ACCEPT button so I'm credited for my help today. Feel free to continue the discussion with me, even after accepting. Thank you and best wishes.