# killall -HUP dnsmasq
That should re-read its configuration.
Then you could try if DNS is working:
# nslookup www.google.com

nslookup should show it connects to 127.0.0.1 and after that it resolves the address. If it doesn't dnsmasq does not know a working outside DNS-server or does not have route to it.

# route
to see routes
and see if there is any /var/run/resolv.conf* files and if, what is inside of them.