The o-rings that are identified as item #8 on the drawing are the most likely culprits. If you remove the spout, there will be one upper and one lower one the barrel section. You can probably find suitable generic replacements at the hardware store. Clean all internal surfaces and apply plumber's silicon grease to the o-rings when reassembling.

Is replacing O-rings something a non-handy person like myself can do?

There are a number of things you can do. This article explains some help for other people who may be having problems

Here is a link that might be useful: How to Fix a Leaky Danze Faucet