Welcome to the glorious kingdom of DE wetshaving! I also suffer from eczema, maar I’m really digging Proraso Refreshing/Green (IT) and Tabac Original (DE) shave soaps. Palmolive cream in a pinch (Kruidvat). I’ve tried so many shave soaps over the past 20+ years, but Proraso and Tabac are great for my use case.

Badgerandblade.com is a great resource to delve into, specifically the Kyle’s Prep write-up.