I would like to break this down in chunks.
EIGRP between sites works when link is up.
-These sites have a DMVPN tunnel between them
As long as the tunnel is up the EIGRP Table will be populated with the networks. When the link is down you should lose these, not right away but eventually the table should be purged. When the link comes back up you should form a new adjacency and the table will be populated again.
What you are saying is the link goes down and only one side shows the EIGRP routes still?
You cannot ping, which should be the case because the one side is down.
Is it the link or the BGP session that is down?
I am a little confused by everything and need to approach this in little bits rather than one large brain dump.
When you say LAN link is down, what do you mean? Is it the switch connection to the router, your host to the switch, etc... Or do you mean the tunnel is down?