1999 Volvo S70 FWD Transmission problem

I hope you’re able to assist me.  We really need some help.
We have a 1999 Volvo S70 FWD.  Recently the vehicle started having issues with
movement after it warms up a little bit.  If the car has been sitting for a period of time,
when its started, the vehicle will engage in any transmission position or gear and does
move the vehicle in the direction indicated.  Within a short period of time (sometimes 2
houses sometimes 2 blocks) the car will no longer move and acts as though it is in neutral.  
I say this because with the vehicle still running if any gear is selected other than park, it
can be physically pushed in either direction.  Once in park the transmission will stay in
park.  It will not move anymore unless the car has been shut down and sits for a while.  

Can I fix this?  What could be wrong?  Is there a video or a step by step I can follow to fix
whatever maybe wrong with this vehicle?

Fluid levels are good, I have had the vehicle in the accessory position on the ignition
switch and have moved the shifter linkage through all the gears making sure to set the
selector in each position.  Placing it back into park and turning the vehicle ignition switch
off (I had hoped this would reset any electrical issues).
Also, what type of transmission is in the vehicle?  I thought it was a 4T65E is this correct?