@rickr I'd put the Meraki switches in first and then look at making the changes.
You don't need a cutover, you can create a new VLAN/subnet for say endpoints, add DHCP to it from the switches, make sure it routes to the existing networks by assigning one access port to it and connecting a computer, then change over each endpoint one by one.
I'd create the voice and endpoint VLANs first and work out what you need to do with the ShoreTel phones to get them passing the right VLAN to the connected device, whilst being on the voice VLAN. It's pretty easy to figure out usually.
Once you have visibility of the client traffic, you may well find out that only some of the endpoints need 1Gb, so don't waste switchports on the others.
As for the servers, it is definitely a good idea to separate them, but also from each other, in application groups. Create a VLAN for the finance servers and have just them in it, create another for a different group etc. You can then use access rules to only allow just the right traffic into that VLAN. Be aware though that you might want to add an MX for access lists (and therefore inter VLAN routing) for those VLANs, as the MS range hasn't got the greatest ACL support.