I recently moved hosting of my domains over from a managed hosting provider of several years to AWS. While there is the inherant time sink of managing the infrastructure yourself, I didn't want to maintain an e-mail server.
My domain registrar, gandi, provides 2 free e-mail addresses per domain, and unlimited forwarding accounts.
While your domain may be hosted on AWS (using aws nameservers), you can update the dns configuration to point to gandi's mail servers.
This is managed in your aws console under Route53, Hosted Zones, under 'yourdomain'. Then create dns record sets as follows:
|TXT||No||10800||"v=spf1 include:_mailcust.gandi.net ?all"||Simple|
..or something like this in your console: