Dating when you have children is a tough one. When I was single I met a nice young lady who worked in the office a few doors down from where I worked. We had a mutual attraction and I decided to see where it might lead. She told me up front she had a daughter, and I had no problem with that.
Our first date was dinner at her home. In retrospect, I think that was a bad idea for a number of reasons. First of all, not a good idea to have someone you really don't know too well coming to your home. Secondly, not a good idea to have your child size someone up before finding out if that person is worth the time or effort.
One of the main reasons I lost interest in this particular lady was a comment she made about her daughter with regards to me. She said her daughter would always be the most important priority in her life. Now, I would have been fine with her saying God was the most important priority in her life. And I could appreciate the bond between someone and their own flesh and blood. But I wasn't too keen on taking a subordinate position of importance in the family planning.
As I'm now a parent myself, I can understand the importance of the relationship between parent and child even more. But there should be a reasonable perspective. Children have a tendency to dislike the idea of sharing their parent with a new partner. And while I think it's important to form a solid bond with the child or children of a potential mate, I don't want to feel like I have to meet their every approval.
I agree with you that the kids shouldn't be exposed to a lot of different people in a carousel of dating. It would leave a bad impression. And I think stressing the special value of a relationship is a wise decision on your part. The idea of not needing someone is a good point to stress as well, especially as you present it in a positive light, as opposed to presenting it out of bitterness like those who feel defeated by their failure to find a suitable mate.
If it is your desire, I pray God will provide you with a Faithful companion.