Thanks, that makes sense. Since this Append is the last line in the function I am not actually trying to re-assign a variable but rather accumulate elements in a new list by applying this function to testList.
Here is the desired output I was looking for using your modifications with a couple additions.
thanks for your help.
checkForAs[aList_,element_,myAccList_]:= Module[{resultValue},
If[Length[StringCases[ToString[aList],element]]>0,
resultValue=aList,
resultValue={}];
Append[myAccList,resultValue]
]
myAccList = {};
testList = {{b,b,b,b,c,b,b},{a,c,b,b,c,c,c},{b,b,b,b,b,c,c,a,b,b,b,c,c},{c,c,c,c,c,b,b},{a,b,b,b,b,c}};
desiredList = Flatten[checkForAs[#,"a",myAccList] & /@ testList,1]
Select[desiredList, UnsameQ[#, {}] &]
Output[574]= {{}, {a, c, b, b, c, c, c}, {b, b, b, b, b, c, c, a, b, b, b, c,
c}, {}, {a, b, b, b, b, c}}
Output[575]= {{a, c, b, b, c, c, c}, {b, b, b, b, b, c, c, a, b, b, b, c, c}, {a,
b, b, b, b, c}}