Have you considered rolling back or sharply limiting your free trials? With the exception of people you've made commitments to, if they aren't accomplishing a business goal for you, you don't owe the world them. There is nothing intrinsic about SaaS that says there has to be a free plan. (Additional options: aggressively using the free plan as a viral spread mechanism. Your free competitors do this, as I learned the other day when on a sales call organized by somebody who would have happily dropped $50 to remove the external branding for that single phone call if they were aware of your option existing.)

We also run on Twilio, and have a 30 day free trial with CC required upfront. While we still have marginal expenses to service free trials in a way that many SaaSes do not, the costs are not terrible and easy to justify as a cost of customer acquisition. (We shoot for, and mostly get, 80%+ margins on the paid plans.)

If you're ever in the mood to chitchat about this, drop me an email. I'd be happy to share about how e.g. moving a bit upmarket and doing a bit of enterprise sales has worked for us.