I was having tuning stability problems and other issues with my American Stratocaster, so I decided to follow the advice of others and got one of these nuts.
I changed the nut along with the tuners and the trem / bridge so I don't know whether just installing this would be enough for you to realise your objective or not - but if you want to do things a step at a time on a budget, I would recommend you start with installing one of these and see how things go.
You could get a tech to deal with this for you, but in my opinion this is easy to install. Make sure you have some abrasive paper and a small file handy to deal with the adjustments you will likely have to make to the nut. You may also have to rub away some width depending on the size of your last nut. But it's easy to work on, and I had no problems.
Once it was on and the guitar was tuned up, the noises that indicate strings are not stable on the nut were miraculously gone and the guitar no longer goes out of tune so easily ! As I said, I upgraded two other key components of the guitar, but I got out of this nut exactly what I expected ! So, I would recommend you get one of these and maybe you'll have the same experience as me !
I plan to install a second one on my other Strat (because I bought two of these at the same time in case I screwed up with the first one - worth considering !! ).