These cars will run without the supercharge belt on the car it might kinda wanna stall out when it first pops off but it will stay running, I was doing about 65-70 when my idler pulley exploded and the engine kinda sputtered a bit.
You know now that I think about it I don't know of to many people with a stock 3800 that have just up and thrown a rod.
Drain the oil and check for metal shavings, I wouldn't jump right to the engine is bad.