Given an algorithm, do worst case time complexity "class" vary with array and linked list?
Note that complexity classes are classes of problems, not classes of algorithms. Specifically, complexity classes are classes of problems that can be solved within particular resource bounds by some algorithm.  So, for example, sorting is in polynomial time because there is at least one algorithm that sorts with this resource bound. (In fact, you probably know many algorithms that do this.)

When it comes to comparing different algorithms for solving the same problem, yes, they may have different resource usage. The fact that you have random access to the elements of an array but not to a linked list means that you can probably design more efficient algorithms based on arrays compared to those based on linked lists. And note that, by using different data structures, they are different algorithms: they involve different computational steps.

Note also that all of this is dependent on your model of computation. For example, if you're solving computational problems using Turing machines, then arrays and linked lists are essentially the same thing, because we can only access the array by moving the tape head sequentially through the elements. Typically, though, when analyzing actual algorithms, we use models of computation, such as the random access Turing machine, that correspond more closely to actual computers.