You are definitely due for a Raptor pet...
Turning it off when you park is PPP,... but I only turn it off, if I'm parking in an enclosed space, or pulling the tank...
...(it will only be a problem, if the floats get stuck,... and I prefer it to leak while parked rather than while running)...
Carb rebuild is easy,... and needs no parts. Just disassemble, clean, and reassemble...
Remove grit from the floatbowl and seat... and run a thin wire through all the passages you can find.. (guitar strings, or welding gauges, work well)... also, spray carb cleaner everywhere, especially on the slide (it can collect a putty-like slime, that interferes with smooth operation)...
...since you have fuel in the filter,.. pull the floats, and shake them, to see if they have any fuel in them... also drop them in a cup of liquid, and watch for bubbles...
3 reasons for fuel in the filter... leaky float, bad float needle or seat,... or bad vacuum petcock... with the Savage, it's usually the pet...
External screws can be a booger... once you get them out, consider replacing with Allen heads...
When doing carb work,.. don't think of removing screws as getting to the job,... removing the screws...
is the job...
Be careful,... and be patient...