It depends solely on your ISP if your actual IP changes upon reconnect - there is no way to force this from the client side. You should be able to see that the connection is coming down an up again in the logs, though.
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Show posts Menuifconfig wgX
wg show wgX
netstat -rn -f inet
tcpdump -ni wgX port 53
and then try a DNS lookup from the client.