My GE corded phone makes a shrill noise when I lift the headset from the base.

Not a problem. Take the base unit with you to Radio Shack and they will get you a proper replacement. (The numbers on the plug in to the unit will say how many volts Direct Current are required to operate is properly and the tag underneath will also state how many milliWatts of power are required at that voltage.) The key is buying the proper plug in end that fits the socket. Radio Shack has units with several socket tips so you may select the proper one. Make sure the polarity of the socket plug you use matches the wire polarity... the stripe on one side of the power cord is the + side on the socket tip.

not a radio communications issue

Unfortunately finding life on another planet would be easier than locating an ancient (electronic years) PLX-500 user guide. However, I happen to have one of those ancient writings (no, it wasn't in a cave at the Dead Sea), albeit a bit tattered and yellow, so I'll try and give you a rundown of it's various features.
The top buttons for headset/handset and mute, and listening volume wheel are pretty self-explanatory. Now let's talk about connecting the PLX-500 to your phone and the other amplifier features.
1) Disconnect the telephone handset cord from telephone base and insert the PLX-500 modular plug into the handset jack on the phone. You then insert the coiled handset cord into the back of the PLX-500 base (there's only one place it can go into).
2) Plug the cord from the headset into the jack in the front of the PLX-500.
3) On the back of the PLX-500 base there are two switches (1-2 and 3-4). These are used to configure your PLX-500 to your phone (most electronic single and multi-line phones, such as Panasonic, will use setting 1 and 4). You will have to configure the PLX-500 to your specific phone. Having connected your PLX-500 to your phone, put on your headset, press the headset button on the base, lift the handset of the phone cradle and listen for a dial tone. Adjust the switch settings until you hear the best dial tone.
4) Remember that your PLX-500 uses batteries located under a cover on the bottom of the base (some phones will work without your PLX-500 needing batteries, check configuration before installing batteries). Speaking of the cover, there is a small plastic screwdriver attached to it that you use for adjusting the volume of your microphone. That adjustment is located on the back of the PLX-500 base next to the configuration switches.
Hopefully this gets your PLX-500 up and running.

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Your solution will depend on what type of phone and headset you have. I have a Polycom 501 IP phone with a headset jack and a GN Netcom 9125. I'm not using the lifter right now, but I seem to recall that an easy way to make it work was to plug the curly cord of the phone's handset into the bottom of the base unit, so that the cord went from the base of the headset to the headset jack on the base of the phone.

First remove the headset cord from the headset jack. That is only used for corded headsets. Plug the short cord into the jack where the curly cord had been plugged into. The curly cord is now going to be plugged into the base of the charger where the picture is of a handset -- the curly cord goes there. There is a A/C adapter that is plugged in to base that is now plugged into the wall.

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Turn over the S11 base and insert the headset slot. This is at the top of the base. Remove the handset cord from your telephone base and plug into the handset jack (there will be a picture). Using the short cord that came with the S11, plug into the line jack on the back of your base and the other end into the slot on your telephone where you removed the handset cord. Plug the headset cord into the headset jack that is on your S11 base. Connect the the A/C power adapter to the S11 A/C jack and a wall outlet. Put on the headset. Press the handset/headset switch and pick up the handset off your telephone, set it aside to make calls. Slide the configuration switch (on the side of the S11 base) until you hear a clear dial tone.

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Please note that your hand receiver has to be off the hook when using the headset.

This is correct, additionally you can use a handset lifter accessory, such as the HL1 or HL10.

Subscribing Headset to Base:

The headset and the base unit are supplied subscribed to each other. However, if you wish to use a replacement headset with the base unit (or if you lose your connection to the base unit), then you must re-subscribe as follows:

1. Return the headset to the charging base.

2. Press and hold both the + and - buttons on the back of the charging base until the red light on the front of the base starts to flash.

3. Let go of both buttons. The red light should continue to flash.

4. Press the mute switch (black wheel on the top of the headset) straight down until a green light appears on the headset, then let go of the wheel, and the light should remain on the headset.

5. Resubscription is finished when the red light on the base is fully illuminated and the green light on the headset disappears.

6. Remove the headset from the charging base lift telephone handset/press headset button if not using a handset lifter. The headset and base should now be connected.

7. If this procedure does not resolve your issue to try a system reset.

NOTE: Before re-subscribing the system make sure the IntelliStand switch (on the right side of the charging base) is in the "_" position and NOT the "=" position.

To Re-Subscribe:

The headset and the base unit are supplied subscribed to each other. However, if you wish to use a replacement headset with the base unit (or if you lose your connection to the base unit), then you must re-subscribe as follows:

1. Return the headset to the charging base.

2. Press and hold both the + and - buttons on the back of the charging base until the red light on the front of the base starts to flash.

3. Let go of both buttons. The red light should continue to flash.

4. Press the mute switch (black wheel on the top of the headset) straight down until a green light appears on the headset, then let go of the wheel, and the light should remain on the headset.

5. Resubscription is finished when the red light on the base is fully illuminated and the green light on the headset disappears.

6. Remove the headset from the charging base lift telephone handset/press headset button if not using a handset lifter. The headset and base should now be connected.

7. If this procedure does not resolve your issue to try a system reset.

Make sure that the power is connected to the headset.  You can tell it's connected because there will be a red solid light on the left side of the front of the charging base.  Check the power cord for any frays and make sure it's plugged into the wall.  If it is, then try the following:
A system reset:
1.  Lift the headset off of the base. 2.  Make sure the headset is turned off.  The green light will not be on on the base. 3.  Hold down the minus button at the same time as the call answer button on the headset.  Hold them down for at least 10 seconds.  An orange light on the headset will flash several times.  It should go off when you let go of the buttons.  If the orange light stays solid, push the minus button once. 4.  Put the headset back in the charging cradle.  Leave it for 30 seconds.  Lift it back out of the cradle and try using it again.

James, plug the phone cord that comes out of the back of the CS55 base into the handset port of the Cisco phone, not the headset port.  Then plug the coily cord from the cisco receiver into the front of the CS55.  Don't use the "Headset" button on your Cisco phone, just lift the receiver off.