These look great! I love your cat, and the green shades are very attractive. The not-perfect prints don't bother me a bit.
A good firm brayer (I like Speedball) is probably the best you'll do without investing in a press. Even though they're referred to as "soft" rubber brayers, they are actually quite firm. I use them for image transfers and collage. Foam brayers would likely not work well.
I've only done a smattering of printmaking in a college class - it was fun, but of course the press makes it easier. (For me, I think doing all the cutouts for this process would be the deal-breaker. But I'm lazy.)
This is very good work! If you think you're going to move in this direction and they're not super expensive, a press would certainly give you more even, consistent results. For occasional fun, I think these are perfectly fine.
Someone with more experience may weigh in here and offer more help.