Question: The full sentence is "There exist 50 different types of sushi, and I have not had one of them."

I understand that the two different interpretations of this sentence are: 

1. There exist 50 different types of sushi, and I have had all but one
of them.
2. There exist 50 different types of sushi, and I have not had any of
them.

How would I write two different formulas of propositional logic? 

The first propositional variable would be:

a: There exist 50 different types of sushi.

Some ideas for other propositional variables are:

* I have had all of them
* I have had only one of them

HOWEVER, the propositional variable needs to be the same for the different formulas of propositional logic. Any ideas?

Answer: 
Define $S$ to be the set of all types of sushi. For any $x\in S$, define $H(x)$ to be the statement "I have had sushi type $x$." (As a side note, you're saying $|S|=50$.)

* "I have not had any of them" $\iff$ $\forall x\in S.\; \neg H(x)$
* "I have had all but one of them" $\iff$ $\exists y\in S.\; \left[\left(\neg H(y)\right)\wedge \left(\forall x\in S-\{y\}.\; H(x)\right)\right]$
First: you really need to use a first-order logic formula for this, not a propositional logic formula.

Now, using:

$S(x): x$ is a type of Sushi

$H(x,y): x$ has had $y$

$i: I$

to say that you haven't had any type of sushi would be:

$$\forall x (S(x) \rightarrow \neg H(i,x))$$

and to say that there is (exactly) one of the types of sushi's you haven;t had you can use:

$$\exists x (S(x) \land \forall y (\neg H(i,x) \leftrightarrow y=x))$$