Select the iPod, so that you see its Summary screen. There is a setting there for Enable disk use. Check that box and Apply. Does that make a difference with the shuffle's light?
NOTE: When disk use is enabled, the shuffle disk remains mounted by the system, even when iTunes is not syncing. To safely disconnect the shuffle, you need to eject it in iTunes. The Eject button is to the right of the shuffle's name.
If this works to charge your shuffle when connected, your new computer may have a "feature" that turns off full power to the USB port when the connected storage device is unmounted by the system. iTunes normally unmounts the shuffle's disk automatically when done syncing (so that you can disconnect it without ejecting). The Enable disk use setting keeps the shuffle's disk mounted (until ejected).
There may be a setting in the computer's Control Panel preferences to disable this "energy saver" feature.