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If you want host your e-mail on office365 as well, then please describe the e-mail issue in more details. for example: Did you get any error message or NDR?
If you want host website on office365 only. Please follow the action plan below to do so. (To P plan and E plan the action plans were different.)
P plan (small business)
1. Verify your domain first.
verify your domain
2. Change domain address. (under admin tab > domains)
3. Find the CNAME of your website under domain manager (under admin tab > domains)
4. Add the CNAME to DNS server.
Add your domain to Office 365 without redirecting your name server records to Office 365
2. Navigate to admin tab > domains > DNS manager > select the domain > select SharePoint online
3. Follow the article below
Use a custom domain name for your SharePoint Online public website address
How are things going on your side? If you need further assistance, please let me know. Thanks.