It's really not that hard to make a list of the payment processors you want to try. First, go through the list and note down the ones that seem interesting. Then, visit each of their websites to check if everything still looks good for you.
Now, you can create a fairly standard email, keeping it under 500 words, where you briefly explain your requirements, the type of business you have, why you want to switch to them, whether you have processing history, etc., and finally ask if they can handle your business.
Once that email is written, it won’t take long to send it out to 15-20 companies and then wait for responses.