Is there a way to automatically send bitcoins from one wallet to another?
You may use /etc/crontab on your Ubuntu:

* * * * * root /usr/local/bin/bitcoin_resender


And script shall looks like:

$ cat /usr/local/bin/bitcoin_resender
#!/bin/bash
if [ $(echo "$(bitcoind getbalance) &gt; 0" | bc) -eq 1 ];then
      bitcoind sendtoaddress YOUR_NEW_ADDRESS $(bitcoind getbalance) "Comment for resender"
fi


It will check every minutes your bitcoind balance and resend each bitcoins to YOUR_NEW_ADDRESS.