It actually depends on the specific B350 model you will get.
So, to sum up, you need a motherboard with 6x SATA (for your 3 HDDs + 1 ODD + 2 SSDs) and at least 1x M.2 socket.
There are about ~10x available B350 mobos that have the above number of ports, these are:
B350 mobos with 6x SATA ports and 1x M.2 socket:
Asus Prime B350M-A (https://www.asus.com/Motherboards/PRIME-B350M-A/)
Asus Prime B350-Plus (https://www.asus.com/us/Motherboards/PRIME-B350-PLUS/)
Asus ROG Strix B350-F Gaming (https://www.asus.com/Motherboards/ROG-STRIX-B350-F-GAMING/)
Gigabyte GA-AB350-Gaming (http://www.gigabyte.us/Motherboard/GA-AB350-Gaming-rev-10#kf)
Gigabyte GA-AB350-Gaming 3 (http://www.gigabyte.us/Motherboard/GA-AB350-Gaming-3-rev-10#kf)
Gigabyte GA-AB350M-Gaming 3 (http://www.gigabyte.us/Motherboard/GA-AB350M-Gaming-3-rev-10#kf)
Gigabyte GA-AB350M-D3H (http://www.gigabyte.us/Motherboard/GA-AB350M-D3H-rev-10#kf)
Gigabyte GA-AB350M-HD3 (http://www.gigabyte.us/Motherboard/GA-AB350M-HD3-rev-10#kf)
B350 mobos with 6x SATA ports and 2x M.2 sockets:
Asrock AB350 Pro4 (http://www.asrock.com/mb/AMD/AB350%20Pro4/)
Asrock Fatal1ty AB350 Gaming K4 (http://www.asrock.com/MB/AMD/Fatal1ty%20AB350%20Gaming%20K4/index.asp)
As to lane-sharing / disabling between the SATA and M.2 sockets, all of the B350 motherboards having just 1x M.2 socket, in one way or another, will disable certain SATA ports (usually 2x SATA ports).
On the other hand, the B350 motherboards having 2x M.2 sockets will allow you to use one of the specific M.2 socket (in PCIe x4-mode, which only disables the 2nd PCIe x16 slot running in x4 mode) and still have all 6x SATA ports enabled.
So, you don't need to get an X370 motherboard as the two Asrock motherboards above (the Asrock AB350 Pro4 and the Asrock Fatal1ty AB350 Gaming K4) would fit your requirements.
Note that in the Asrock mobos with two M.2 sockets, the socket M2_1 only runs in PCIe3.0 x4 speeds (not SATA) and disables the 2nd PCIe x16 slot (the one running in x4, not the x16 top slot for the GPU). Using the M2_1 socket does not disable any of the 6x SATA ports. The socket M2_2 only runs in SATAIII speeds (not PCIe) and disables one SATA port (the SATA3_3).
Reference:
AB350 Pro4 manual:
http://asrock.pc.cdn.bitgravity.com/Manual/AB350%20Pro4.pdf
Fatal1ty AB350 Gaming K4 manual:
http://asrock.pc.cdn.bitgravity.com/Manual/Fatal1ty%20AB350%20Gaming%20K4.pdf