NETDOM worked, sort of.
I could not understand why I couldn't get NETDOM to execute in the Command Prompt, so I uninstalled/reinstalled RSAT. This time, I was able to get NETDOM.exe to run, but I was not successful in resetting the password. This is what I saw:
Microsoft Windows [Version 6.1.7601]
Copyright (c) 2009 Microsoft Corporation. All rights reserved.
C:\Windows\system32>netdom.exe resetpwd /s:server /ud:domain\user /pd:*
Type the password associated with the domain user:
The machine account password for the local machine could not be reset.
The network path was not found.
The command failed to complete successfully.
Now I know that there's a great possibility that the reason this didn't work is because I do not have the PC on the local network like it usually is. I took it home with me and attempted this while establishing a VPN connection to the domain controller. I thought it might work since I can login fine while off the network, but I get the same Trust error message when trying to login while connected to the VPN.
I'm heading off to return this PC to the office, where I hope it will be able to successfully reestablish the trust.