This should be pretty simple. All you need to do is configure your DNS records so your site works from the AWS server and your emails work from Namecheap.
You should do this in the panel to which the domain is currently pointing. If you use cPanel/WHM on Namecheap I'll recommend you to point your domain to their service and configure the DNS records from the cPanel interface as it's usually more friendly.
In general terms what you'll need to do is:
Configure an A record for your-domain.com pointing to the IP in AWS and a CNAME record for www.your-domain.com pointing to your-domain.com. This will make your site to work from AWS.
Then, configure an A record for mail.your-domain.com pointing to the IP provided by Namecheap (where your emails are hosted) and then configure the MX record so it points to mail.your-domain.com. This will make your emails work from Namecheap.
Take into account that this is a general solution, your email provider may have special MX records so you should get in touch with them so they can help you out.