I'm using an int variable:
month = dp.getMonth() + 1;
currently getting an output of "2" and when I do the following:
if (month<10){
                month = '0'+month;
            };
I get: 50.
Your problem is that your '0' char is being coerced to an integer. Since '0' has an ASCII  value of 48, you're getting 48 + 2 = 50.
Note that what you're trying to do won't work - you can't add a leading 0 to month, as month is a number. A leading zero only makes sense in a string representation of a number.
As explained in this answer, here's how to produce a zero-padded number:
String.format("%02d", month);   
 
    
    