The syncing issue you mentioned actually happens in many different programs and is not necessarily OBS specific. But now to your question:
You might wanna check my Spreadsheet on the mostly used Capture devices.
https://docs.google.com/spreadsheet/ccc ... GNXRkpDLWc
Overall the Avermedia devices (PCI-Express ones) are known to work quite well. So for example the
Live Gamer HD (up to 1080p60 input with 1080p30 or 720p60 output)
Game Broadcaster HD (up to 1080p60 input with 1080p30 output or 720p30)
or AverTV CaptureHD (up to 1080i30 input with 1080i output or 720i30 (720p30 output on same input)
They will soon get an USB device aswell that looks to be good, after release well know more.
Black Magic is the next Big and renown company for capture devices of all kind.
They offer an PCI-E and USB version of the Intensity series. But they are known to be a bit tricky when it comes to input and output resolutions. Maximum input is 1080i30 as far as i know.
Then of course there are some very expensive capture cards, above 500$, but I guess you look for a consumer device :)
I can personally not recommend using one of the several USB boxes that came out in the last years. They are mainly ment for recording, not for streaming.
Greetings Jack!