You need to break this down into the local (connected) networks, that is,
the subnet that the NIC are in and remote networks, that is, networks that
are reachable via ,say, the default gateway. My first guess is the the
default route is flipping back and forth as each NIC gets its address
renewal....

So when you next have connectivity problems, first check that you can ping
the IP address of another system on the same subnet as the NIC, and do this
for both NICs. I'm guess that will work. Then try pinging a remote system
and see how that correlates to where the default route is pointing, using
netstat -rn.