I use Whatsapp in that exact scenario that you are describing. It's an old Android phone connected to wi-fi and the phone is never touched for anything outside of Whatsapp.
Having it setup that way, or for you with FB/IG, if there is no SIM in the phone, do not install anything else, no contacts in phone, you can effectively use it anonymously.
I suggest you use an Android phone so you can download APK files to install, disable Play Store and similar network services, and maybe once a month download the lastest APK and install to update it. I do that for Whatsapp or updating since I have all Google services disabled on the phone.