Hi,
How are you discharging your power supply components? Improper discharging can cause electric shock, burning, and component damage. People often wear a pair of insulated gloves, and discharge the relavent nodes accross a 10MOhm (or so) high power resistor. Make sure you leave sufficent time for current drain. Some people have hooked up LED's to sense when this is complete, but it's really up to you, so long as you know you and your equipment are safe.
When you talk about handling your live transformer with bare skin, I am somewhat concerned for your safety. Are you sure you know how to handle live electrical equipment? There are often exposed mains terminals accessible to your touch, within power supplies. I really wouldn't suggest servicing live equipment, unless you really know what you are doing, and it's really necessary. If you absolutely must, then you should use proper safety equipment to do so.
For your power supply, it may be easier, and even cheaper, to replace the whole thing, than to replace the transformer.
The lighter power supply is a switch mode supply. Transformers use Faraday's Law, and a ratio of wire turns, to accomplish a voltage step down. Switch modes use frequent switching, and low pass filtering to average the voltage output. Switch mode supplies are used a lot these days, but I seem to recall reading that the A500's switch mode supplies were less reliable, and had lower output than their alternative. Not sure if that's true, just thought I'd mention it.
As a side note, a lot of old transformers actually used rust as the insulation between the laminations, as it was so cheap and easy to coat the steel plates, and the rust would also bond the plates together. However, this was not very good for efficiency, or longevity. I don't know when this was phased out, though I expect it's not current practice.
Some transformers actually whine/hum when they are new. You can see this in some CRT TV's. Aging monitors also suffer from this.
