Way too many Linksys Routers to pinpoint the one you have, anyways, do you have the latest firmware for your Router?
If not, ensure you get it, you may have to reset Router after updating
As your connection is flucuating, is there a lot of interference between the Router and Wireless adapter
Cordless phones on same frequency, Microwaves, etc..
Have you tried changing broadcast channel on Router?
What encryption are you using?
Router is not only broadcasting N, it must also be doing B/G as you are getting a wireless connection
Interesting enough, did you get this wireless adapter for free? Why not go for a Wireless N adapter?