Would baking a cake for 2 hours at 350 degrees F be safe to eat if it's still moist? I'm trying to modify a recipe for a cake which calls for 3 cups of shredded butter nut squash. I thought I could replace this with roughly the same amount of pureed pumpkin (like what you would get in a jar).

However, the cake is still moist inside, after more than doubling the baking time it originally called for.

I don't mind that the consistency is too wet, I'm only concerned about cooking the eggs enough. Unfortunately, I don't have a thermometer to test the internal temperature. Is it safe to assume that at 350 degrees F, with the cake baking for more than 2 hours, that it reached the internal temperature of 138 degrees?

The cake is about 1.5 inches in height. I baked it for 1:10 minutes, then an extra 1 hour with tin foil on top (to prevent the top from burning). The top and bottom of the cake are not moist at all.
Unless you are very young, very old, or immunocompromised in some way (chemotherapy, leukemia, HIV, etc), slightly undercooked eggs are fine.