There are two things that I suggest. I have a 90 XJS V-12 conv. and had a similar problem.
Remove the black plastic cover over the relays just behind and above the battery. There are two relays for the fuel pump located there. Pull the red diagonal stripe relay and see if the problem continues overnight or over the period that the car "sleeps". I ended up installing an additional relay because I was getting some type of electrical bleed through that relay that was slowly motoring over the fuel pump. You can confirm if this is the problem by using mechanics stethascope and placing the rod against the fuel pump. I am not sure if the fuel pump was relocated into the fuel tank by the 94 model year or not. On the 90 it is mounted under the battery.
If it turns out that you can still hear a low growl from the fuel pump with the ignition off, I will tell you how to install this additional relay that opens the wire causing the problem when the ignition is off. It took me months to find this problem. My battery was going dead overnight. I installed a relay to open that wire and the problem stopped.
The other thing to try is to pull the fuse for the power antenna . Removing the relay won't solve it. The short is somewhere "upstream" of the antenna.
A friend with a 92 six cylinder coupe found that their short was in the trailer wiring circuit. They pulled the fuse and the problem stopped. For their 92, something besides the antenna was on that circuit. You can determine which fuse to pull from the fuse chart in the owner's manual. On the 90, the antenna is on the same circuit as the trailer "caravan" wiring.
See if either of those is your source.