First of all, please accept my condolences for your loss. A beloved spouse, with whom you built a family, can never be replaced.
That said, BE”H, you can find much happiness – and bring happiness to another at this phase of life, creating new memories and a new “family” feeling. I applaud your sensitivity for concern about how to best prepare your children for this new relationship, as even if they are fully grown and independent, a new marraige will affect them in many ways.
The first thing is to have a family discussion that you are feeling ready to explore dating again, and to reassure them that anyone you would marry would also need to appreciate your relationship with them. Ideally, before you make any decisions (which are yours alone to make ultimately), it would be kind to take things slowly and for them to have a chance to meet her and get used to the idea.
One of the big reasons for negativity of children surrounding a senior parent’s remarriage, is that surrounding inheritance. It would be good to consider how your remarriage may impact their potential inheritance and make some plans, according to what you deem appropriate. It may make sense to draft a pre-nuptial agreement, as anti-romantic as that sounds, it’s also reasonable for those who have significant assets.
Regarding housing and relocation, i think that is wholly individual and must be decided in the context of the potential relationship. If relocation to another area is not an option, just make that clear prior to embarking on another relationship.
I wish you the best of luck finding a warm and sympatico companion to share life’s blessings with. 🙂