Here's a twitter thread with a good summary of the current situation, as well as some great ideas and resources for taking action. These include a link to a list of Health Legislative Assistants for various senators, and the suggestion to reach out via social media to friends in states with key senators who need to be influenced:
It also points out that, worryingly, staffers report calls to senators' offices have fallen back to near to "normal", pre-Trump levels.
This Trumpcare Toolkit also provides a list of the key senators who need to be influenced and gives some easy ways to reach them. There's also a link to a letter asking the Senate to take expert and public testimony on the bill:
Edit: Another list of staffers dealing with healthcare for Republican senators:
From the responses to that thread: if your senators are Democrats, you can call and ask them to hold off on all non-essential business until a public hearing is scheduled for the new bill.