Do I have to run a full node to keep a personal/offline wallet?

You can use mymonero.com webwallet if you prefer. This is much simpler and faster (at the cost of being a webwallet so, although particularly secure, you are still delegating a bit). Read Is there a webwallet for Monero? What is MyMonero?

If you prefer to use a "regular" wallet, you do not even need to run a full node anymore. The lightwallet unofficial wallet does this for you. Of course, here too, you are delegating a bit. Read Where is the GUI wallet?

If you wish to run a full node, note that at the present state of development, running a full node & using the wallet is not a particularly friendly venture, so it is wise to really go through the setup instructions available and see if your computer skills will tolerate the effort. Experienced/Advanced users will have no trouble setting up a node & wallet, although it still requires a lot of RAM, until the conversion to a database is complete.