Is it possible to Clear all variables (but not functions)?

Automatic way based on Names

Here is a simple modification of the recent answer of @R.M, which is based on the definiton of a variable as a symbol which has an OwnValue defined:

Clear @@ 
   Select[
      Names["Global`*"], 
      ToExpression[#, StandardForm, 
         Function[sym, OwnValues[sym] =!= {}, HoldAll]
      ] &
   ]

If your code is in some context other than Global` (which is what I'd recommend), you can put that context in place of Global` above.

Automation based on custom assignment operator

As an alternative, and IMO more systematic way, you may define a custom assignment operator which you'd use when defining a variable:

ClearAll[defVar, $variables, clearVars];

$variables = Hold[];

SetAttributes[defVar, HoldAll];
defVar[def : ((Set | SetDelayed)[lhs_Symbol, rhs_])] :=
  With[{result = def},
    $variables = Append[$variables, Unevaluated[lhs]];
    result];

clearVars[] := $variables /. Hold[vars__] :> Clear[vars];

Now, you use this as

myFunA[int_Integer] := int + 1
myFunB[int_Integer] := int*2
myFunC[int_Integer] := int - 3

defVar[myVarA = 1];
defVar[myVarB = 2];
defVar[myVarC = 3];

And then,

clearVars[]

A digression: evaluation leak in AppendTo

Interestingly, in the line $variables = Append[$variables, Unevaluated[lhs]], you can not use AppendTo instead, at least not in the form one would expect it to work. The reason is rather subtle: AppendTo seems to be implemented in the top-level code, but more importantly, it leaks evaluation:

vars = vars1 = Hold[];
a=1;
AppendTo[vars,Unevaluated[a]];
AppendTo[vars1,Unevaluated@Unevaluated[a]];
{vars,vars1}

(* {Hold[1],Hold[a]} *)

You see that an extra layer of Unevaluated is necessary to counter the evaluation leak. I would think this is a wrong behavior for AppendTo, but can not state with definiteness that this is a bug.

Remarks

Which way to choose depends on the situation. The first method is fully automatic and will work best if you keep all your work in a specific context distinct from Global`. The second way requires more discipline but works regardless of the contexts you are using.