Assuming you do in fact have a cellular connection...
Are you accessing a 3G signal?
Skype works over WiFi. Have you tried when connected to WiFI?
I only signed in once - back when webOS was still supported and just switch it on and off as needed. I know there are problems with some email accounts in that 'historical' ones work with some patches, but new accounts can't be authorised. This has also been solved recently with various updates. I don't know anything about the Skype mechanism, but perhaps it works in a similar way and the OSSL / Certificates update will fix the problem or maybe there is a new authorisation process. As Skype is a proprietary system, patching this might not be possible.
Again, I don't know if it's relevant, but is there any reason not to upgrade to webOS 2.2.4?
There are a few here with spare devices, so perhaps they can report on their Skype experiences after doctoring - if they are able to log back in without issue, then there is hope for you.