I can assume the fuse blows while going from Drive to Park as there is no solenoid noise when pressing on the brake while turning my car off.
I tend to doubt it could be the shift lock solenoid itself. It's not connected directly to the fuse, and doesn't act like a switch, opening and closing a circuit that could short the fuse.
Are you sure that, when you stop and move the lever to P, the fuse isn't already blown?
The fuse also powers the stop lights, so when the fuse is blown, the lights won't work. If I understand the description correctly, you can bring the car to a full stop using the brake pedal, as is normal, but only when moving the lever from D through N, and R to Park, it appears that the fuse blows and you can't get the lever to P. If you park at night where you can watch the stop lights through the rear view meter, can you see the stop lights go on when the brake pedal is pressed, and then go out as the lever is moved, while not changing anything else (foot still on the brake pedal)?
The most common cause of the stop light fuse blowing is a short in the stop light circuit. Often there's a bad dual filament bulb (stop and tail), melted or flattened contacts on a dual filament bulb, or a fault in a socket (corrosion, conductive trace). It's possible that a new fuse will work initially so that the lever can be moved from P to D, but with repeated braking it fails, so when you go to move the lever to P and press on the brake pedal, you don't hear the solenoid click, and the lever can't be moved to P.