Splitting your message into two parts has a couple of benefits. First, it makes you more memorable. Second, you seem less rehearsed. If you're reciting from a script, you're probably not going to forget a key component. Prospects will automatically trust you more.
When you leave a voicemail do not use your full name because that immediately raises a red flat that you are a salesperson making a cold call. Instead, say your first name only followed by your company name. This will make your customer feel that you are much more familiar with each other than you really are. Saying your last name would defeat the whole purpose of demonstrating familiarity.

There is a possible chance that your WiFi is the reason that you are unable to set up your greetings for voicemail. Many users solved their problem by changing their network from WiFi to Cellular data. Your carrier provides the cellular data and that can be a better choice for voicemail settings.
It’s a good practice for each of your team members to have their own personal business phone numbers. There are many reasons why they shouldn’t use their personal cell phone number for business, so you’ll want to give them their own phone number through your VoIP provider or phone system.