Godaddy DNS redirect to AWS, screws up my mail server...

As a long-term solution, though it is not simple, I suggest using dynamic DNS through one of the services such as dynu.  Amazon can then switch you to a different server or different IP address around every hour if they feel like it;  you're still online.  Same goes for moving your site to a different hosting provider; you can be up and running in ten minutes on a new site with no DNS setup.

• Register with a dynamic DNS service.  I use dynu; there are others.  Paid is not necessarily better than free.  Set both www.domainname.tld and domainname.tld
• At your domain registrar, set your domain's DNS A record to point at the dynu servers.
• Install ddclient on the linux host.
• Configure ddclient on the linux host with the domain name, username and password at dynu.
• Restart the linux host or start the ddclient service.
• Within 30 minutes the DNS address should percolate through to the TLD servers.

Note that when A and MX records resolve to different IP addresses, some email trouble can be expected.  Not a great deal of trouble, but a little.  A few MTAs (possibly due to misconfiguration) appear to ignore MX records and send to the target sitename instead.

The web site requires an A record or a CNAME record.
Assuming that your domain is called "example.com", what you may have today are MX records pointing back to your root domain.  

example.com  A  123.123.123.123
example.com   MX   example.com

What you have to do:
a. Start an AWS instance
b. Get an elastic IP and associate it with the instance. Lets assume that you got 54.55.56.57
c. Create a new A record in Godaddy DNS, call it "mailserver.example.com" and point it at 123.123.123.123
d. Change your MX to point at mailserver.example.com
e. Change your apex A record to point at 54.55.56.57

You need an elastic IP because by nature of the cloud, you will want to change instances and keep a stable, constant IP address.

Old MTAs had problems with different MX and apex A records, but frankly, last time I saw something like that was in 2001. I have worked with literally thousands of domains in the past years, and never had any problems.

I did this, but when typing in the domain, it just goes to a white screen. nothing else

Can you please share the domain?

bncicomputers.com