For my intro health economics class, my professor asked us to explain how certain scenarios impact the supply and demand of health care (using only supply/demand analysis). He raised an example of a pandemic resulting in widespread illness affecting both healthcare workers and the general public. To start off, I thought that there would be an increase in demand (more consumers entering the market) as well as a decrease in supply (fewer suppliers of health care). Theoretically, prices would increase and it'd be difficult to figure out how quantity changes. However, just logically and thinking about COVID, it would make sense that there is a shortage because quantity demanded would surpass quantity supplied given that suppliers of health care are now demanders. I'm struggling with how to represent this graphically where shifts in supply/demand lead to changes in equilibrium price and quantity, but simultaneously result in a shortage. Is my thought process correct, and is there a way to explain supply/demand shifts with a concurrent shortage?
This thinking is not completely correct. You seem to be confused about quantity demanded and demand and quantity supplied and supply.
Demand is the whole relationship between price and quantity demanded. An example of demand is not a number such as 100 units of hospital beds at day $t$ (even though media often incorrectly use word demand in such way), demand is a relationship between price an quantity. That is demand would be some $Q(p)$ for example $Q(p)= p-100$.
So if demand shifts that does not necessarily mean quantity demanded changed. Indeed demand might increase and supply might decrease but quantity demanded could stay exactly as it was before the shock if price adjusts.
However, in real life prices do not adjust instantaneously. Prices are sticky for various reasons. Walmart won't increase prices of toilet paper immediately when shoppers start emptying the store. Similarly hospital won't just raise prices instantaneously when there are too many people in ER. Price adjustments usually take times as they cant be just done by nurses/doctors at ER who have the immediate information about demand, there are laws and regulation prohibiting it, issues like menu costs etc.
In principle even if supply is extremely limited and demand extremely high, there will be some large enough price that will make sure that quantity demanded is equal to the limited supply despite enormous demand. In principle there is always price at which quantity demanded drops to zero for some extremely high price. Prices do not have theoretical upper bound. But in real life there are price rigidities that simply don't allow for instantaneous adjustment. Moreover, there is also 'anti-gouging' legislation in many countries so it might be even illegal to rise prices too much at a time. For example, in the US there is 'Price Gouging Prevention Act of 2022'. Especially when we talk about medical care the prices are tightly regulated almost everywhere in the world, no matter whether we talk about medicine or procedures. Even in countries where there isn't a strict price regulation (e.g. US) there are other laws such as the above mentioned act, and beside regulation there are natural reasons for price rigidities such as above mentioned menu costs and so on.
Hence the way how you model this is as a simultaneous shift of supply and demand assuming prices are fixed at below equilibrium level. You can visualise it as follows: