I haven't followed your sitch carefully, but how often do you really need to email or call?
Yes, 2-3min weekly during handoffs is enough for most co-parenting business. “Johnny began sniffling two days ago, I’m giving him 150mg ibuprofen and checking his temperature every 4hrs, and he has a violin recital from page 31 due Friday.” Every 3-6mo something comes up like school registration or a medical cost or grades suggesting a quick call or e-mail.