Could someone provide insights into the implications of a hypothetical Schrems III for EU-based SaaS companies that host their servers in the US, particularly those containing Personally Identifiable Information (PII) like email addresses? Essentially, would Schrems III mean that we'd need to immediately move our servers to EU soil, or risk fines?

Whether your servers are in the US or not, if you do business in the EU, EU rules apply. It might be that you will legally not be able to offer your services in the EU, if you have servers in the US, because those can the accessed by US authorities at any time, without you even learning about it. It is probably safer to have servers in the EU, if you want to do business in the EU. Servers in the EU not provided by any US hoster, since that hoster is vulnerable to being ordered in the US to transfer data from EU to the US.