Does adding t as a argument work for you?

extractSymbols[ equation_ , t_] :=
      Block[{varsAndCosVars, deDupedAndAll},
       varsAndCosVars = (Variables @ equation) /. (Cos[vars_] :> Variables @ vars);
       deDupedAndAll = 
    DeleteCases[DeleteDuplicates[Flatten[varsAndCosVars]], t]];

If you want an "operator form," you could define it thus:

 extractSymbols[t_][ equation_ ] := ...

Then extractSymbols[t] @ equation.

Okay, several things going on here. First, from the documentation for Needs (emphasis added)

Needs["context`"->"alias`"] adds "alias`"->"context`" to $ContextAliases, providing the ability to use a shorter or easier to type alias to access symbols in that context.

Contexts loaded with this form are not added to $ContextPath.

Unless you have some specific need that you haven't explained yet, you will want to have the $ContextPath updated so that your new package context will get searched. This is how you will avoid prefixing all of your new symbols with the full context. So, just do this:

Needs["ContextA`" , FileNameJoin[{NotebookDirectory[], "ContextA.wl"}]]

Now, we need to get your new symbols into the ContextA` context. The way to do that is to expose them after BeginPackage["ContextA`"] and before Begin["`Private`"].

Read the documentation for BeginPackage and Begin. It explains how each affects the context and context path.

So, you could do something as simple as this:

extractSymbols::usage = "Extracts variables from models of a particular form, minus the t variable";
pSort::usage = "symbol precidence, with sub-sort on component ID";
M;w;A;t;

Now, get rid of all instances of "a`" (your alias), and your tests pass.

You should show us how exactly you set up your paclet (e.g. how you used BeginPackage, Begin, or whatever). But I suspect that the problem is to do with your symbols M, w, A, and t. You should declare those symbols in your package context. pSort and extraceSymbols should also be declared, but based on the way you asked the question I assume that you already declared those.