Classic/Antique Car Repair: 1970 rear differential on a corvette 350/300hp stock, metal fillings, rear differential

Question
I AM GETTING A HUMMING SOUND FROM THE REAR, ONLY WHEN I LET OFF ON THE ACCELERATOR. I;VE TAKEN THE FILLER PLUG OFF TO CHECK THE OIL, WITH MY FINGER I REMOVED OIL, ITS THICK LIKE HONEY, VERY DARK AND SMELLED BURNT. I DID NOT SEE ANY METAL FILLINGS IN THE OIL.SHOULD I PUMP THE OIL OUT AND ADD NEW OIL? SHOULD I GET IT CHECKED OUT INSTEAD? DO I CONTINUE TO DRIVE IT UNTIL IT BREAKS DOWN OR WILL IT COST LESS TO REPAIR IF CHECKED ASAP. WHAT SHOULD I DO!  

Answer
Jimmy;
Hea- well first of NO-NO do not wait for it to blow- dam it might just be a U-joint and if it blew in the front end and stuck into a pot hole it could flip the car over, you do not want to wait for that to hapen do you ?  it could be a bearing,or in the gears, or a adj. I would take out all of the oil oil and fill it with new H.D. gear oil and fill it till it runs out the hole. than see how it is, if no good I would take it in so fast as it might be a small thing,but if it blows it could turn into a super big bill. you do not want that right.... Good Luck..