u/Happy-Replacement886

I've been reading as many threads as I can, but I haven't found my answer. My kids play youth baseball (Little League and Babe Ruth) and youth hockey. We are heading into High School age and I thought if I'm going to try to get a better lens, I may as well do it now before this season of life is over. I have an a6000 and upgrading the camera body is not an option. I'm currently using the silver 18-200 and it is not sufficient. I can get the shots, but everything is a little fuzzy and I can't get much as far as background blur. I'm looking for a unicorn option -- not too heavy, not too big, and not too expensive. I am also coming to realize that there may not be a single lens to take on both of these sports. My kids pitch, play infield, and one is a goalie. So I don't feel that I need a super long zoom. Lenses on my current radar:

* Sony 85mm f1.8 prime (good for hockey rinks by the glass? Maybe not long enough, but feels like a good investment none-the-less. I have the 35mm prime.)

* Sony 70-350 G (good for baseball but probably won't do the trick in the rink, have seen plenty of negative reviews

* Tamron 70-180 g2. Could this decently do both?

* Tamron 50-300. Could this one be good for baseball and leave me some room to buy something else for hockey?

* Samyang 135mm f1.8. Longer than the 85, maybe an alternative?

The Sony 70-200 is way too heavy (I'm small female) and I'm really just a mom taking photos that I wish were better. I'd feel like it's overkill.

Thoughts? Thanks for the help!