Finite Automata with succinct representation of chains of states
This is a very partial answer, but I have some ideas:


Clearly the union of NSAs can be taken without any blowup - just use the nondeterministic union of the initial states.
As for determinization, you'll have a double-exponential blowup. Consider the language $L_k=(a+b)^*a(a+b)^k$. That is, the $k$-th before last letter is $a$. You can easily construct an NSA of size $O(\log k)$ for it, but any DSA would need to "remember" the last $k-1$ letters (the same argument that works for DFAs would work here). So a DSA would need $2^{k-1}$ states, which is a double exponential blowup.
As for union of DSAs, this is more of a conjecture: consider the language $L_p=(a^p)^*$ for a prime number $p$. This can be recognized by a DSA with 2 states and $p-1$ written on one of the transitions. Now, consider the union $L_p\cup L_q$ for $p\neq q$. It seems to me that you'll need a lot of states (i.e. $p\cdot q$) in order to recognize this with a DSA. But I haven't thought this through completely.