Hi
Try ipconfig /flushdns to reset the computer DNS cache. Make sure the computer request the MX record from the correct DNS server you updated too.
Can I ask if it cause issue ? I ask as if it's only your local network it mean that MX is technicaly not used at all, unless you have a third part mail server inside your LAN that could use it. (which I will doubth if you are on O365 already)
The MX entry is used only when a server send an email. The computer that are in sync with 0365 send via Outlook(MAPI) to the remote MS server, which handle the send somewhere else, or in the same tenant.
The important MX is for the external domain, thus where everyone else business check to send you an email.