I’m using an exchange wallet without experiencing any problem here, maybe its because the amount of money is just small.
Just stop using an exchange wallet when you withdraw but for deposits, I think that you won't experience any problem from there. Based on my experience, it's not really a good one and the exchange might surprise you with an email noticing what you did and where your source came from.
Well, to be more safe I have to think for other options as well and many here suggested Electrum, I’m not sure though if it will work.
Electrum is very well one of the best wallets out there, easy to use and lite.
How about if you’re going to withdraw and deposit thru LTC? Any good wallet to use for?
Just the usual multi-currency wallets will do, search what you think is good for you as a choice.