Not experienced in events in Japan myself.
But there are two different types of handshakes. One are the theater handshakes, this you get tickets for by buying the theater edition of a single, and each single gives you 1 handshake ticket. This one is done before a single releases and involves using a Japanese address (and card, probably) to put in your picks for your handshake. It's kinda complicated, and I'm not sure how you do it. Another type of handshake is the nationwide handshake event, where you get tickets from buying the Ltd. version of a single that comes with the handshake tickets. Then you just go to the venue when your girl is slated to have a slot, and queue. They have concerts from time-to-time, but I have no idea about the timings for those. But April is usually concert season, so you might catch one.
For now though, I recommend you apply for super enpou for all the shows happening during that time period, as well as apply for enpou when the date is closer. 3 weeks is a decently long time, so you might win a show. If you have a railpass for the duration, then you can register for NMB too.