Yes, a new adapter (assuming its fairly new manufacture and not just "new" to you) would likely be able to pick up the faster signal from the new router. Its just your a/b/g/n descriptionthat sounds odd to me. Maybe you are picking up the 5 GHz band on the wireless G , which is nice. I thought N (which is definitely faster than G) was required.
anyway , glad you now have the screaming fast wireless. It does help a lot for just internet use and as you say , for video its make or break for HD. Although my G is working fine , I do see a new dual band N router in my near future. Its like a new fishing rod. Do you need it? probably no. Do you want it? Always yes. Will you buy it? Probably yes.
