Is this ECC based messaging method secure?
You should consider “sign-then-encrypt” instead of “encrypt-then-sign”. 

Note that it is very important that the “encrypt” in “sign-then-encrypt” should be authenticated encryption. If you don’t use authenticated** encryption, you should definitely use an athenticating “sign-then-encrypt-then-MAC” approach.

Besides that:, when thinking about all this, you should definitely keep ECIES and ECDSA in mind.