I assume the camera casing is getting warm to the touch after a few minutes?
Do you have an old AV input on a tv or something? If so connect the camera video out to that instead (but keep the battery connected) that way you can eliminate the video input part of the vtx (I've had this where the receiver only gets a black screen and the camera video out works fine, meaning something is broke in the vtx on the input side, it's quiet a common problem on those tiny KingKong VTXs).
Also please check the pins inside the camera, make sure they are ok and you didn't accidentally try and plug in the cable the other way and flatten them.
(I just noticed I have exactly the same FPV setup as you, camera and vtx, so yeah, separate them both and check them individually ef ore you go tearing the sparrow apart.)