Wired & wireless together

I am managing a small (4 desktop computers plus one used as a server) network. We are using a wired network thru a switch, which also connects to our FIOS modem/router.
The problem is that our lan traffic is VERY slow.
I suspect that since all 4 computers are always busy on the internet as well as the lan, that the internet traffic is slowing things down since everything shares the switch.
Can I put wireless network cards into the 4 computers and run all internet traffic wirelessly while running all lan traffic thru the wired switch?
If not, is there a better solution?

You could use a separate wireless router with wireless adapters at each PC, but it would be easier to install a new wired router that would route LAN traffic to the existing router and all other traffic to a new Internet connection.