Okay. You know I am in the midst of what you are talking about.
I don't think you do stop caring about the other person and why do you have to in order to have another relationship. Friends for an example, you have many friends and on different levels. Does that mean you must stop caring about one in order to introduce a new one into your life? No. why should it be any different?
Just because someone leaves you and wants to be separated from you, doesn't mean you all of a sudden stop caring about or loving that person.
You know that I do struggle at times with my own morals and what it says about me and to others because I did get involved with someone so fast and so early. There is a lot to deal with at times and even when you think all is okay and that have dealt with issues you won't truly know until you are involved with someone else. Things come as you go along and then you can choose to handle them or not. You will not know what those issues will be as long as you are not putting yourself out there.
I think it is okay for you to be interested in dating but if you do feel that uncomfortable with the idea of being legally married and doing so, then you are going to have to make the choice to end it.