Do I really need a Static IP to fix my NAT type?
Not usually.
A dynamic IP vs static IP is not relevant to NAT. As long as you have a public IP address that doesn't change from moment to moment. For most ISPs they change your IP address rarely, like once every few days or only when you reboot your modem.
NAT (Network Address Translation) is the system in your router that allows packets sent to your public IP to make their way to a specific IP address on your LAN. In the case of game consoles, this would allow you to be the host of a game if your NAT is considered to be "Open". (Probably type B or A for Nintendo).
Your router and/or modem may have firewalls and port forwarding rules that are blocking certain kinds of connections to or from the Nintendo Switch. It is also possible that your ISP has a firewall or blocked ports on their network too.
If all of the firewalls and NAT devices between the Nintendo servers and your Switch are configured properly, you should be able to achieve a NAT type of "A" even with a dynamic IP.