I think that (especially with families) scheduling is everything. He shouldn't be trying to plan a way to slip in a quick date with her unless it is part of the schedule. Especially if it is during a time that is dedicated to family (which I am slightly unclear about - was he trying to find a time THAT WEEK or after you got back?). Especially when it is HIS parents that you're spending the week with. A certain amount of spontaneity is nice in relationships - being able to decide to go out to dinner or a movie or squeezing in lunch when it wasn't expected is great, but it can't take away from other obligations (in this case - spending time with parents, daughter, and you). Also, your relationship should get as much focus time as the one he has with his new interest. If they get a date night every week, it's not unreasonable AT ALL to ask for a date night too (even if it can't be a going out date night due to lack of babysitter - you two can do something special at home and treat it like a date). He has to spend time nurturing the relationship you two have, his family, AND his new relationship.
It seems like your discussions haven't really been discussions, either. You tell him you have a problem with x, he says "okay, I'll stop" and it's over. While that's all well and good for you, is he feeling like he's being heard? Is he afraid that if he responds with more of a discussion it will cause an argument? I always worry when people just roll over and give me what I want. lol
On that note, though, he really does seem to be trying to make you happy. He didn't push to cancel your date night (I'm hoping because he realizes it's important and rare). It is soooo easy for texting to get out of control. I fall prey to that vice ALL THE TIME. I have to put my phone out of sight/hearing range or I am tempted to look at it and respond every time it goes off.
The texting thing is also difficult because you live with him. You see each other a lot. You recognize that date nights for the two of you are rare, but do you realize that the times for idle chitchat are rare for him and her since he doesn't see her often? Also, what do you consider "your time?" Is it anytime you two are home together, or do you have set evenings/times every day that are "yours?" Asking him to only text when you're not around or otherwise engaged is a bit much, I would think. Asking him not to text during dinner, your date nights, specific family-oriented time (movies, games, and other activities you do with your daughter), and such is more than acceptable since he really does need to be focused and 100% there for those types of things, but during the down time if you're just sitting around relaxing or each doing your own thing, isn't it okay for him to text? Even if you notice it?