When I accelerate, it sounds like I am running.

Can you make the noise occur by holding the brakes and raising engine speed in gear while standing still? If so, have someone look at things like the fan blade to see if it's hitting the shroud when the engine rocks. If it will occur in neutral when just raising engine speed, listen to the power steering pump, idler pulleys, and generator to see if the noise can be located. Temporarily removing the serpentine belt will tell if it's coming from one of those components.

There is a tool you might be able to borrow or rent from an auto parts store that borrows them called the "Chassis Ear". It is a set of six microphones, a switch box, and headphones. You clip the microphones to suspect points, then drive around while listening with the headphones. You can move the microphones around to zero in on the source of the noise. Be aware that many mechanics have never seen or even heard of this tool. Suspension and alignment mechanics use it to find rattles, squeaks, and other noises.