You only have the option for the Per-call. If you are making a call and do not want it to pick-up there is a number you call to stop it - But just for that call. That was ONE of the reasons I stopped it.
Since the Verizon Voicemail will answer calls to you number when you are using your phone, a second incoming call to your number while your own answering machine is taking a message WILL BE forwarded the the Verizon Voicemail system --possibly giving you recorded messages in 2 places. Note that each increase in the ring count will add 6 seconds to the amount of time it takes to forward the call to the Verizon Voicemail system.
If you really don't what the Verizon Voicemail system which can handle multiple calls simultanteously (something you own answering machine cannot do), your only option is to phone Verizon's business office and request that they remove the Verizon Voicemail service from your line.