A TRACON Controller reported an NMAC event during a heavy traffic period. Controller stated being short staffed during these heavy traffic periods contributed to the event.

Narrative

During the Holiday time; the ZZZ and ZZZ1 airports get a lot more traffic from the box haulers. The amount of traffic in and out of ZZZ1 has risen exponentially. I can barely work it sometimes; which is why I'm filling out this report. I had a lot going on. Before we get into the airplanes; on my previous session working a combined position because we were; you guessed it short staffed; I had a pilot want to file an NMAC. Now; I did what I could by calling traffic and I feel I prevented a collision from happening. But I didn't use the good ole' 'traffic alert' phraseology and I was concerned I was going to get in trouble for that. So that was weighing on my mind. Let's add I'm tired of working overtime (yesterday) for xx years now; I'm getting old; and I dream every night of retiring. So the airplanes. We're having our 'evening rush' on crack since it's the holidays and everyone myself included buys everything on the internet. I approved a stack from one ZZZ Controller. Of course they really helped me out by making the first guy super high and super-fast; but what's new? Aircraft X wanted practice approaches; to an airport that I wasn't servicing; but the aircraft had to get across my finals to get to where they were going. That honestly took way too much of my focus. So the Aircraft Y was southbound northeast of ZZZ. In front of them was a Aircraft Z or something like that. One of the larger ones. I had the Aircraft Y get the heavy in sight and had them follow. The issue with this was the Aircraft Y was keeping it high above the wake. I get it. I really do. But they were just a little higher than I anticipated they would be so that Aircraft X and the Aircraft Y were a conflict. It wasn't a 'loss of separation' kind of conflict but one that used too much of my attention and I didn't notice the horrible vector for the downwind I gave the aircraft into ZZZ1. I saw it when it was too late. I didn't even bother calling traffic. I just vectored the aircraft away. Sadly; it sent me into a issue and I had the worst session I've had in over 10 years. Just terrible ATC. I feel awful. This is hard. All this ZZZ1 stuff is HARD. So many delays. Maybe I'm just too old. Flow. I don't know. The amount of ZZZ1 arrivals and departures is too high.