Short answer is yes, I am relatively certain that you must still contact the airport manager prior to flying within 5 miles of the airport. Though they're somewhat ambiguous, the FAA still has the "you must call" terminology in places on their website and they reference flying by "community standards". Always better to be safe than sorry so I say when in doubt, call. The airport manager may tell you there is no need but then you'll know for sure.
Keep in mind that some Class E airspace starts at 700 or 1200 AGL. Areas like this is where Class G underlies the Class E. Hence you can fly but must call. However, your non towered airport may be Class E2. Does it have instrument approach? Class E2 starts at the surface and requires authorization through AirMap, DroneZone, etc. The UAS Facilities Map and VFR Map along with chart supplements will help you ID the subclass of Class E airspace. If you don't see the LAANC overlay covering any given controlled airspace airport on the facilities map, then you must request authorization via DroneZone.
Also note that when using AirMap, you get an additional set of overlays with more details if you highlight the optional "AirMap Recommended Guidelines" option.