The transmission of my 1995 riveria seems to be slipping

Not sure about the noise load liter but with a 1995 I'd start with the transmission it's probably starting to slip.
You can change the fluid see if that helps if so you'll need a new one soon.

If you are driving on the highway in 4wd drive the poise may be from the tyres trying to slip and equalise the drive line torque that is building up . Rule of thumb for all 4 wd vehicles is to restrict 4wd to loose surfaces (gravel-snow-ice -grass- mud --any surface that allows the tyres to slip) to relieve the torque wind up.

Sound as if you may have a transmission problem

Hello Matt Panek,

Any chance you have a POLICE chopper following your every move??

Whirring ... Seriously? (might mean you overdosed).

Could be a loose fender liner... or maybe a wheel coming off and EVERYTHING in between (wheel bearings or dragging a kid's tricycle).

When everything is bolted down, clear and working fine ... there is nothing finer than the freedom of driving your car...

But when they start making strange noises...
They can be QUITE DEADLY.

A part coming loose... or even a plastic bag sucked into an air intake can have fatal consequences if ignored.

If you cannot immediately find the cause of a trouble...

Muffler shops, mechanics & service stations would love to have you visit...

Don't risk SAFETY... when all the FINE FOLKS are
just waiting to help (sell lots of things)...

Listen to what they are telling you...

A loose muffler bracket can save your entire exhaust system if addressed early... but that same MUFFLER CAN KILL
PEOPLE in the car behind you if it FALLS OFF at freeway speeds.

Be safe... Always check out & resolve NOISES before
endangering yourself or others

Carnac the Magnificent

make sure the axle joint didn't fail then on the pass side of the car behind the axle there is a cover that houses the differential open that and check the gears in there

SEEMS LIKE TRANSMISSION FLUID LOW.TRANSMISSION SLIPPING TRYING TO GO IN DRIVE.CHECK TRANSMISSION FLUID LEVEL. IF LOW ADD FLUID UNTIL YOU REACH FULL MARK ON TRANSMISSION DIP STICK. WITH ENGINE RUNNING A IN OPERATING TEMPERATURE.IF FLUID LEVEL OKAY.CHANGE TRANSMISSION FLUID AND FILTER IF YOU HAVE NOT CHANGED IN A LONG WHILE.A DIRTY STOPPED UP FILTER WILL CAUSE TRANSMISSION TO SLIP.CANT PICK UP FLUID.

Your linkage could be loose; low on transmission fluid; or you need a new flywheel

Check transmission fluid level, then look at transmission linkage. Some are in a place where they can be bumped around and knocked out of place. Tweek the shifter a little on both sides of the resting position when you shift. If the noise goes away. It is probably linkage related. If you can sample the tranny fluid, check for burnt smell and or metal flakes. Aamco will check it free but dont let them sell you repairs as preventative measures. They will try.

Disconnecting the drive shaft is a bit more of a tricky prospect than you were told. Odds are it won't eliminate the whirring, either. First, just disconnecting it from the system is going to leave a gap in your transmission that allows for all kinds of crud to get in, and may even (in certain transmissions) allow for gears to drop out of place and potentially cause damage. There are kits available that allow you to remove the shaft and place a "dummy end" in the transmission. Another possibility is to leave the end of the shaft that inserts into the transfer case in place, while only disconnecting the U joint in the shaft and removing the remainder. Problem is, either one of those is still going to have a spindle and, if in fact that is the cause of your sound, you'll still get that noise.

You'll be better off either putting up with the noise and leaving things alone or getting a more accurate determination of the exact cause and deciding if it's worth fixing.

I take it this is a manual transmission. Drain some of the oil out of the bottom of the transmission, and look at it to see if there are any pieces of metal in it. The drain plug should have a magnet. This should be all under warranty.