I don't have a closed form and I doubt you'll find a nice one, but here's a recurrence relation that would allow you to compute the number of admissible strings by recursion.
Let the numbers of identical letters in the string be $n_1,\dotsc,n_k$. Consider what happens if you remove one instance of the $k$-th letter from an admissible string. The remaining string has $n_1,\dotsc,n_k-1$ letters, and it is either an admissible string or an inadmissible string with two identical letters adjacent. In the latter case, we can contract the two adjacent letters into one to get an admissible string. Thus, we can count the number of admissible strings like this:
where the factor $n_k$ accounts for the fact that we can remove any of the $n_k$ instances of the $k$-th letter, the factor $\sum_i n_i$ accounts for the fact that we can remove it from anywhere in the string and the factor $n_i-1$ accounts for the fact that we can split up any of the $n_i-1$ instances of the $i$-th letter. The initial values are $a(n_1,\dotsc,-1,\dotsc,n_k)=0$ and $a(0,\dotsc,0)=1$.
For your example, this yields
where in each step the recursion was used to reduce the last non-zero letter count.