I know bitcoin core is the most popular/suggested software to run a full node, but are there other options? I've heard of others like Bitcoin Unlimited. My question is, would those nodes running alternative software still have to have that software adhere to the protocol rules of bitcoin in order to be a full node on the network?
No anyone can desgin their own.Its just that rules and coding knowledge is required.Armory wallet is the best alternative to bitcoin i can recall
you are talking about writing code for a full node which consists of a lot of different components most of which are security sensitive like the consensus rules enforced by the network, wallet related parts such as signing,... this is not an easy task to perform!
as for Armory, it is only an addition on top of bitcoin core so you can't say it is a different software.