Hello Timothy,
he EWS protocol has built in limit for request length so maybe the larger messages that you send hit the limit. Please try to adjust the server settings and make sure that the setting “httpRuntime maxRequestLength” is set to 20000 for 20MB limit or to 50000 to 50MB limit.
If this does not help, we will need full EWS logs.
As for the downloading for offline use - this setting is present only for IMAP accounts. Exchange accounts download all content as default.