Does anyone know what this houseplant is? Is it a Dieffenbachia? I'm debating if I should repost it to a bigger container. I think it needs a lot with better drainage.

Yes, your plant is a Dieffenbachias (Dieffenbachia) but repotting to a larger pot will only cause additional issues to what appears to be an already stressed plant. The current pot size looks fine and depending on the size of the plants root mass, it may even be a tad larger than necessary. Ceramic type pots do not dry as quickly as the plastic or terra cotta types and the soil appears a bit on the wet side. I'd advise adjusting the watering routine to allow the soil to become a bit dryer between watering.

Thank you very much!