I’m using the app GPXLogger on android. Simple, no bells and whistles. A note about battery drain: normally when you have the phone’s location services on, it’s also trying to use the signal from surrounding cell towers to determine location, and if you’re in an area with inconsistent or weak cell service, the phone’s searching for service can eat a lot of battery (may also be messing with the GPS accuracy you’re seeing). That’s why I use a GPS app but put the phone in “airplane mode;” no more searching for cell service, it just relies on the gps chip which shouldn’t use much battery. I notice huge differences in battery consumption due to this, probably a 10x increase in rate of consumption with cell service on.
Syncing later using the photo timestamps is a pretty easy process that multiple free software programs can do for you. Here’s a tutorial thread with a lot of discussion on different software and hardware combinations and the processes of different users: https://forum.inaturalist.org/t/geotagging-photos/66/22