How to check if DFA is actually a correct model of the intended system? Suppose I have modeled a deterministic finite automaton of my system. How can I check if the traces generated by this system, actually represent the model I had in mind? 

For example, say I have DFA A which models a vending machine. To ensure I did not model any wrong transitions, is coming up with  (safety/liveness) specifications and performing verification the only method I can follow to ensure my model is correct? Many thanks.
If you want a guarantee, then yes, verification is the only approach.  That's pretty much the definition of what verification means.

If you want a heuristic check, there are a number of testing methods based on synthesizing test cases (perhaps based on the DFA) and then checking that the vending machine's actual behavior on those test cases matches your DFA's.