I bought this last week and the headset feels like it's fitted properly, but it doesn't look like it…

Cups aren't in properly, they should be pressed right into the frame.
Having said that, I don't know what the internal of a Cannondale headtube is like

the headtube may need reaming to allow the full insert.

That's what I suspected, but is there something about headshock I don't know?

should it have an integrated headset?

It's highly unlikely that someone will manufacture a headset that's outwith standards i.e. won't fit.

Get it pressed in in lbs.

Might be obvious, but have you used a headset press to fit the cups? Looks like they would go in with the right tool? Though dont do anything drastic!! Oh and dont ride it too far or too fast like that…

I didn't do it, I bought it like that. To be honest I didn't spot it but it was such a good deal I would have bought it anyway.

Allen keys are out… i'm going in!

How old's the frame – I've an 2004 frame that will accept 1.5" headsets with no problem – I believe that earlier frames had a shallower rebate in the headtube that won't accept a 1.5" headset cup without modification.

Cannondale head tubes are machined to fit a short insertion headset anything that needs to go in further wont fit.

that's not a headshock is it? and I'm not sure you'll get far with allen keys 😉

I believe that earlier frames had a shallower rebate in the headtube that won't accept a 1.5" headset cup without modification.

Cannondale head tubes are machined to fit a short insertion headset anything that needs to go in further wont fit.

This sounds plausible, I think it's a 02/03

You need to ream out the headtube to a deeper depth to accept this headset or buy the correct headset for a cannondale headtube:
e.g.
Cannondale compatible 1.5" headset
This is described as follows:
"Additionally, we make a Double Xc-short, which allows the use of a 1-1/8” fork with Cannondale HeadShok frames."
The reamed depth on cannondale headtubes is only about 10mm (to accept headshok bearings) hence the need for a short insertion headset.

Or…cut the headset you have down.

Although it may work fine as it is.