We have used ConstantContact off and on for clients for close to 8 years. Their metrics reporting is far from ideal, and I do not like that they place their branding at the bottom of your newsletter. That said… they are reasonably priced for my clients, and they are a trusted email provider. Meaning, emails sent through them are in most cases not immediately marked as spam, as can happen when you use some other ESPs (email service providers).
In your specific case, your recipient’s email systems are accustomed to receiving the eNewsletter from ConstantContact’s IP addresses, basically whitelisting your eNewsletter. Horror stories can develop when you move from a trusted ESP to a new one, especially an ESP with a shaky history and clients that do use them to send spam.
As for leveling off, that’s natural. Radical drops, though, and I would think that you have, in fact, been blacklisted by some of your users.
“David offers an incredible amount of diversity in developing marketing and sales programs, and has the hallmark of great managers - he asks questions and probes before devising a winning strategy.”
~ Business Manager, Miele 
David, this is an excellent point. It’s the old “throw the baby out with the bathwater” issue. This seems somewhat reactionary, and the results of changing ESPs at this point may have a HUGE impact on their program.