How to tell when bitcoin has really stopped running
On linux: 

ps aux | grep bitcoind


Or even:

watch ps aux | grep bitcoind


But to answer your question slightly differently I'd suggest keeping existing bitcoind running.  Then your new node add a connect statement with the local ip of the old node in the bitcoin.conf file (located usually in $HOME/.bitcoin/).  

Your new node will then solely connect to the existing node and download, via your local network all the blocks from the existing node.

Also if both PCs are imobile you could leave it that way to avoid duplicate downloads from the internet.  An alternative if the second is a laptop for example is to use -addnode instead. From here:

-addnode=&lt;ip&gt;          Add a node to connect to and attempt to keep the connection open
-connect=&lt;ip&gt;          Connect only to the specified node(s)
-dns                   Allow DNS lookups for -addnode, -seednode and -connect (default: 1)