Hello Fambrough,
According to the description above, I understand that your domain name registrar is Network Solution, DNS hosing provider is Host Excellence, and you want to add the domain to Office 365 while keeping the current public website. But currently the Network Solutions does not allow you to change any DNS records in case of losing data.
In addition to the suggestion of mchv2.0, the following steps are for your reference:
1.Change Name Server records back to the domain name registrar Network Solutions.
2.Move your email to Office 365 and keep your current public-facing website
3.Verify your domain and change name servers
Regards,
Cherry Wang