Originally my dryer had burned the plug on the power cord and the receptacle. I was told it could be a bad cord so I replaced it and it worked fine for several months. Now It has fried the wiring in the junction box in the house wiring about 6 feet from the dryer. Do I have some type of bearing binding in the motor or elsewhere or does someone have any other suggestion as to why it is doing this?

If your bearing was binding, I'd bet it would be "singing" also.
Is this an electric dryer? If so, maybe the wiring is not sized for the appliance?

It is an electric dryer and it did have a vey faint squeek.

What I suspect is that the first time the cord & plug had burned was caused by a loose connection in the receptacle, probably the plug was loose or the blades of the receptacle were loose.

When you replaced the cord the receptacle should have been replaced, since there was a hidden problem inside. Now the cord had burned again because of this hidden poor connection inside the receptacle.

Replace both the cord/plug and the receptacle. Assuming this is a standard electric dryer, the circuit should be rated a minimum of 30 amps, to include a 30 amp 2 pole breaker, #10 copper wire, and a 30 amp 250 volt rated receptacle and cord.

We have a 40 amp 2 pole breaker, I'm not sure what size the wire is, and a 30 amp 250 volt rated receptacle and cord. They also have the central air running from this same circuit, should this be on a separate circuit?

Yes both appliances need their own circuit. How is the AC outlet connected? If they connect at the dryer receptacle, I bet this is the cause of the problem.

The ac is split off at the junction box where it appears the dryer wiring gave out. The ac is still operating.

Then all there is are defective connections in the dryer outlet.

I can redo the connections myself but we don't have enough space in our breaker panel for another dual throw breaker, what will have to be done to run another circuit so both the ac and the dryer have their own? Until this is done is it safe to run the ac and the dryer as long as we don't run both at the same time?

Don't just re-do the connections, replace the dryer outlet and the damaged cable also.

A new circuit will need a breaker, outlet and cable installed for either the dryer or AC. If 30 amps then a 2 pole 30 ampCB, #10 copper cable, and the appropriate outlet.

There should be no problem running the AC without the dryer. I thought the dryer is out of commission due to the fried cable and outlet and cord?

I can get the cord, receptacle and connection taken care of myself, I just need a professioinal to run the new circuit. So I can run each seperatly until I get someone in to set-up my new circuit.

Yes , but what happened to "Now It has fried the wiring in the junction box in the house wiring about 6 feet from the dryer."

I will pull new #10 copper wire from the junction box through the conduit to the dryer outlet.

I agree with TK in regards ro putting these on their own circuits.
If you have no room in your panel, have you considered a gas dryer?

We called in the professionals. They found that the connections had gone bad due to age and aluminum wiring. They cut all the wires back to good cable and applied anticorrosion paste to the aluminum to keep them from reacting with the copper and coming loose again. They ran a check on the load with everything running and said it was at the maximum for this circuit, but we are still going to check into running another line so both the a.c. and the dryer have their own.