If your 8600 is not compatible with your VoiceMail service then there is nothing you can do. You would have to scrap your telephone answering service or the printer and purchase another in hopes of it being more compatible. If not then you need to choose either faxing or answering machine features and move on.
Not sure if you have explored this but there is a feature on HP fax machines called Telephone Answering Machine (TAM). When TAM is enabled the printer waits for the answering machine to pick up the line and monitors the connection. If fax tones are heard then the fax machine takes over from the asnwering service and takes the call. If not then the fax machine lets the VoiceMail take over. When using TAM you need your fax machine to have its Rings to Answer higher than your Answering Machine service. That way the Answering machine always picks up first.
Otherwise you are going to have to look into either getting 2nd dedicated fax line for your fax machine or scrap the physical fax and go with an Online Faxing service. Both have thier advantages depending on what you are comfortable with but Online Faxing is by far the most relyable way to fax in todays digital phone envirionment.