I didn't have trouble saying "no thanks" to alcohol at events after getting sober. Often, that was that, and no-one would ask why.
Initially, if someone did ask why (because I was a known drinker in my social and work circles), I usually got away with "I'm taking a break from the booze". In my drinking circles, people did that from time to time too. "Detoxes" were pretty common in that group.
As months went by and it became clear this was a long-term change, I would say a little more, like: I felt much better not drinking, that I was tired of using it as a crutch or stress-reliever, even sometimes stating that there's a history of alcohol abuse in my family and I thought I should stop before it became a real problem.
About a year on, sometimes I even said straight out I had a problem with booze and I don't drink anymore. It just got easier and easier to be straightforward, because the shame factor (as an alcoholic) diminished and the pride factor (as a recovering alcoholic) increased.
Do what works for you and what you're comfortable with. It can help to be a little evasive early on IME, because you're just dealing with so much. As you get more comfortable in your recovery, it becomes less and less of a big deal, and your responses will probably get more confident.