Q: What is the best way to externally measure improvement in my customers’ experiences?

First is good that you have a good CRM plataform where you can do a performanced and deep analysis of yours customers.

Then you´ll need to choose wich avaliation metrics you´ll be using along the measure time, use these metrics as benchmark to work.

1) Measure expectations for each customer support channel

2) Measure response times in the several possible ways where you communicate with yours customers (email/chat/phone...)

3) Measure customer sactisfaction after each interaction (after a survey for example)

4) Measure multiple overall customers sactisfaction after purchases

There is a several customer loyalty and customer effort to do business with your company  measurements.

The best way is to deal with a mystery shopping agency, set the criteria that you want to measure and they will do the visits to your shop and get back to you with detailed reports

You must have to understand that customer's problem is mine problem, then go forward and resolve the customer issue.