How do you know if a language accepted by a DFA is $Σ^*$?
You don't input a language to a DFA. Each time you run the machine, the input is just a string. The language accepted by the DFA is the set of inputs that cause it to end in an accepting state (called a final state by some authors). Therefore, the automaton accepts $\Sigma^*$ if every possible input leads to an accepting state.

So, to decide if an automaton accepts $\Sigma^*$, you need to check if every input does indeed lead to an accepting state.