Update: Since writing this I learned about a much better way to send Dogecoin transactions than using Multidoge. Apparently the optimal wrote to signing and broadcasting transactions is to use a minimal wallet client like Coinb.in, then you could just delete Multidoge. Coinb.in is pretty technical to use, but if you can't get Multidoge to sync it's worth looking into.

For the past few weeks I had a couple old Multidoge wallets that I couldn't send funds from because the wallets would never finish syncing. The problem, it seems, was that Multidoge was failing to establish or maintain a connection with an active node. Here are the steps I took to get Multidoge to finish syncing:

1. Make sure Multidoge is completely closed. (It must be closed or the changes you make in the following steps to the multidoge.properties file will be overwritten when the app closes.)

2. Open your multidoge.properties file in a text editor. By default this file should be located on Windows at C:\Users(your user name)\AppData\Roaming\MultiDoge. On Mac, it should be located at (your user name)\Library\Application Support\MultiDoge. (Note: on Mac, the Library folder is hidden by default. If you can't find it, follow these steps).

3. Goto the Doge Network Peers page and copy an IP address near your country.

4. Back in the multidoge.properties file, find the addnode=### and peers=### lines, and replace the IP address after the = sign for those lines with the IP address you copied, then save the file. I've seen some recommend adding many comma separated IPs to the peers line, like this peers=98.144.228.249,79.66.188.193,109.202.19.190 and so on.

5. Start Multidoge. If the connection is successful, you should start seeing Synchronizing with network in the Messages tab (Menu Bar -> View -> Messages) followed by the date of the block currently being synced. It may take several minutes before it begins syncing.

While this solution has worked for me on both Windows and Mac, it does not work 100% of the time. I've found that I have to try a few different nodes before it works, and even then it takes hours to sync the whole blockchain.

If you have any questions, or know of a better approach to fixing the Multidoge syncing issue, please comment below. Thanks to madmarcel for pointing me to this solution. Good luck!