UPDATE: The only way I've found to be able to get into the game WITH the DLC and play normally follows:
Start Origin
Start ME3
"Press Any Key to Continue"
Let ME3 connect to EA servers
AFTER the first flash of the "Checking DLC", but BEFORE the second flash - Disconnect from the internet
Wait until the game tells me that I have lost my connection to EA servers
Click OK
Alt-Tab out of the game
Reconnect to the internet
Log back in to Origin
Go back into ME3 (by clicking on the minimized taskbar icon)
Click the button in the lower right area of the screen to "Connect to EA Servers"
Play the game normally.
However, after playing, I have to disconnect from the internet BEFORE I leave the multiplayer menu to return to the main menu....otherwise the game will crash again. My system specs are listed farther up in a spoiler. If someone would like me to repost them or some other informaton, let me know.