|I'm guessing your setup is as follows:|
Internet >> Modem >> ASA >> Switch >> Clients
Please confirm that, or explain if it's setup different.
Your tests seem to have ruled out everything on the external of the ASA unit, I would suspect the issue is with it. I've never used one of those appliances and couldn't begin to help you troubleshoot it but would suggest you contact Cisco if you have a support contract with them. Or the provider you bought it from if you have a contract with them.
If you don't have a support contract with anybody, and nobody here can help you, I would suggest finding yourself a cisco forum or two and asking there.
It matters not how straight the gate,
How charged with punishments the scroll,
I am the master of my fate;
I am the captain of my soul.