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Its a tough one, as they're going to have opinions and you should take those on board. My children are my world, so their opinions matter more than anything to me. I'm not ready for the dating scene right now, but if i were, they would definitely be involved. I also think its a good idea to keep the respective 'potential partner' well away from your children until you've established a friendship with that person and have gotten to know them some. If and when you feel the relationship is heading in the serious direction then and there should you talk about meeting on mutual ground and not at your place of residence, so that your children will get to 'feel' this person out without feeling over burdened or put out by him being in their environment/safe zone. All the best. x
I think you'd be wrong to limit your choice of new partners based on your DD's anxiety, especially as she can't really tell you why she feels that way. But if you do meet someone, you'll need to handle it carefully and ensure she doesn't feel pushed out, either by him or by any DCs he has. As you say, it's probably some way down the line you'd be introducing her to anyone, and she may have changed by then.
I'm lucky that my DCs have always been quite keen on me having a partner, and were keen to meet my DP's DCs. But I'm lucky that he didn't pay too much attention to his own DS1's hostile views at first, or he'd never have got together with me (his DS1 said I was an evil witch, and wouldn't speak to me at first, but gets on fine with me now).
Actually getting to know real people is quite a different thing from thinking about it in the abstract, so I'd just reassure her that she's always your speical DD, but that you might someday have other people in your life too. After all she may have DCs herself some day, and a mother-in-law and who knows who else - and none of that will mean she's not your DD.