Is there a lightwave 9.3 update? I cant find it. the 9.0 version wont work. 
If you own LW 9.x then you own LW 9.6 which is the latest (non 10 version that is - you need to pay upgrade to get 10).
Go into your newtek dashboard at register.newtek.com and see your “my downloads” section. In my “my downloads” 9.0, 9.1,9.3 and 9.6 are available.
Lightwave’s FBX exporter in 9.6 works fine with unity. I use that one (with it’s settings being the 2009fbx)- don’t know about 9.3 nor why you would want to use 9.3 specficially, since 9.6 does everything 9.3 does and has bug fixes. Try that one instead maybe… 
I cant find my serial number, so I cant register. Any idea where it would be?
Well it might be on some sales receipt or email you got when you bought it.
In LW modeler
Help> About Lightwave Modeler
Brings up a panel that has your Lock ID. You could use this with customer service to sort things out.
You really need to write that stuff down and keep it in a known place.
There is a spot to write it in the paper manuals if you got them.