Saying you need to charge your watch after 3-4 runs doesn't really mean anything. Are your runs 20min 5k's or are you running 4x 3hr long runs? Specs say this watch gets "up to" 7 days in smartwatch mode and "up to" 13 hours in GPS mode. In my experience, Garmin's estimates are WAY optimistic (unlike Apple who are the ONE electronics mfg who are honest in their battery estimates). If Garmin says 7 day in watch mode, you're probably looking at 4-5 days max assuming ZERO GPS usage. Expect this value to fluctuate depending on how far away your watch is from your phone. If you turn off your phone at night or deactivate Bluetooth due to airplane mode, then your watch will burn extra battery constantly searching for your phone's Bluetooth. If you've configured your watch's backlight to Always On or Raise-to-Wake, then that also burns battery which Garmin's estimates don't account for. Also, 13hr est. for GPS is way optimistic. Expect to burn around 10-12% battery per hr of GPS activity in actual usage. In challenging condition, battery drain will be even higher.
Regarding your Q... is GPS still on when you "exit out" of runs and go to the main screen. It depends. "Exit out" has several colloquial meanings that don't correspond to actual Garmin button press functions so I don't know what you actually mean.
When you intentionally start a GPS activity, the watch REMAINS in active gps mode until you INTENTIONALLY END that gps activity by pressing the START/STOP BUTTON (upper right button) AND THEN cycle thru the START/STOP MENU options and intentionally select one of the options: Resume, Save, Resume Later, Discard. You MUST select one of these options in order to END a gps activity and deactivate the gps receiver.
If you intentionally start a GPS activity and then simply cycle through the screens until you see the date/time screen (without first ending the activity), then the GPS receiver is STILL RUNNING and burning battery.