It wasn't that Bucky was actively avoiding Steve. He wasn't at all. He just needed time, as he'd said. Harboring feelings for someone for the better portion of a century (because even under HYDRA's control, those feelings were there and the whole reason Steve had even been able to break that mind control twice) didn't exactly make it easy to just let it go. What he had to figure out wasn't necessarily that he should let it go, but rather, he had to evolve. The feelings could still exist, just had to be refocused and transitioned into loving Steve as his best friend.
When they saw each other, he would offer a little smile but they still had yet to actually talk much since the day Steve had come over and they'd broken up, for lack of better words. Moving forward had been hard at first; sometimes it felt impossible. But he'd been doing it. Being in a relationship with Clint and Natasha had been a big part of that. Them being in his life allowed him to feel other things besides the extreme sadness, the loneliness, and the overall feeling of being lost when what used to be one of the few constants in his life no longer could be there at all; not in that capacity anyway.
That evening, he'd been outside his house hanging out with the little Pokemon he'd brought home with him. They'd both bonded over being someone who people kind of shied away from because they looked grumpy most of the time but underneath it all actually had soft hearts and could be very loving when they felt so inclined. As soon as Bucky had seen how other Pokemon ran away from the little pink dog he'd immediately decided he needed to take him home. They'd been bonding ever since.
A figure running nearby had caught his eye and of course, it was Steve. Steve, running with what looked like some kind of dog. So of course, Snubbull started to run toward it because he wanted a friend. "Whoa, wait--" But it was too late and Bucky had to run after his companion which took him right in the path of Steve. The pink dog started jabbering in the way Pokemon tended to do, as Bucky had learned, so Bucky looked at Steve, a little unsure. "Sorry, he-- He keeps trying to make new friends."