I have access to create A .... records
And that's what you need to do. You need to create an
A record on the subdomain that points to the "other hosting package". Specifically, the
A record contains the IP address of the server that hosts your website.
If you change the NAMESERVERS (ie. the server that manages your domain's DNS) then it will no longer be managed by Office365.
Ordinarily, if you simply have a domain and a website host and everything is managed by the website host (DNS, website, email, etc.) then you would change the NAMESERVERS to point to your website host. But that is not your situation.
Alternatively, you do change the NAMESERVERS to point to your "other host" BUT you would then need to manually recreate all the DNS entries at the "other host" (the other DNS provider) in order to configure Office365. (Incidentally, you would create the required DNS entries at the new host before you changed the NAMESERVERS, otherwise you would experience downtime.) There will be approximately 6+ DNS entries you would need to recreate, depending on the Office365 services you currently use. The exact entries required is explained within the Office365 control panel. (It is probably easier to create a single
A record instead - as mentioned above.)