I'm a security guard, not an artist, but after 25 years on the job I've seen a lot of people with hobbies, including guys who draw. Here's what I picked up from their conversations - in Photoshop for portraits you need to use brushes with soft edges and low hardness, like a regular round soft brush at 30-50% opacity. The thing is, it's not really about the brush itself, it's how you set it up. Try playing around with the settings - lower the hardness to 0%, turn on size dynamics based on pen pressure if you've got a graphics tablet. Here's an example: when I go out for a run in the morning, I don't immediately pick up speed, same thing here - you need smoothness in how you apply the tool.
As for ready-made brushes from outside sources, yeah it makes sense to download a set of portrait brushes, like on sites such as CGTrader or Gumroad there's plenty of decent options, often even free. But I don't think you need to switch to different software. Photoshop handles it fine, the standard set is just limited. One more tip - check out the Texture option on brushes, add some canvas or paper texture, that'll immediately make everything look more natural and alive, I'm telling you.