The part that would have irritated me the most is him texting with her while you were ring shopping. If it was my husband or my boyfriend I would have been angry at them. It's not even about them texting their other partner. It's the fact that he wasn't completely present with me. My husband is addicted to his smart phone. He loves constantly checking Facebook, his email, games he plays with friends, etc. I've talked to him about how constantly being on his phone when he's with me makes me feel like he's not focused on me and doesn't really care about what we're doing. And if we were wedding ring shopping, it would offend me even more. We're doing this thing that is supposed to be all about us and he isn't completely present.
I don't have any advice other than making certain times just about you and him, no phones allowed. I hope it helps though to know that someone else would feel irritated about it too.