Amongst the mass of junk chairs on Amazon there are some under $300 choices that are considered bearable, notably the Mellet (sold under various brands including here and here) and the Ticova Ergonomic Chair here. At worst if you don't like one you can take them apart and ship them back to Amazon for a refund.
You can certainly get usable chairs for under $300 on Amazon - the Clatina Mellet (Costway version here) or the Ticova Ergonomic chair here for examples - but these would not in any way be an upgrade from an Amia that is in good working condition. Quality office chairs have gotten very expensive in the last few years.
$300 can get you something used but it can be hard to find something you like in good condition. Old Herman Miller Aerons of a similar age to your Amia can sometimes be found in your price range for example.
This has been asked several times and no one has ever found a difference. Go with the cheaper one. The only other generally recommended alternative is the Clatina Mellet, which you can get for $220.
BTOD does not recommend the Ticova for people 6’2 and up, but the Mellet is suitable at and above that height as long as you grab a replacement cylinder (as I had to do), or if your height is more in your torso rather than your legs.
Not knowing your height, I would recommend starting with the Ticova, and returning it if it doesn’t work for you. Mellet tries a few more advanced things but kind of falls flat on the details. Ticova is a bit more basic but seems to have fewer surprises for people.
edit: Also pay attention to the weight ratings for these chairs, which is lower than with most of the really high end manufacturers.
Either of the chairs below will likely be better for you. People generally complain about the Hyken’s mesh seat sagging soon after purchase.
https://www.amazon.com/Ticova-Ergonomic-Office-Chair-Adjustable/dp/B0B9YG2LJN be sure to apply the $80 coupon just below the price listing
https://www.amazon.com/COSTWAY-Ergonomic-Armrests-Adjustable-Breathable/dp/B0B17ZRT6N this one is just over your budget, but it comes highly recommended as best in class in this price range by numerous sources
One benefit of Amazon is the easy return process, so if you don’t like one of these you aren’t stuck.
You have so many options for that price point.
HON Ignition 2.0 $400 https://www.amazon.com/HON-Ignition-Mid-Back-Adjustable-Lumbar/dp/B06Y3PGPR2
Ayalon $400 from Staples or maybe must cheaper elsewhere if they’re legit https://www.staples.com/staples-ayalon-mesh-and-fabric-task-chair-black/product_2722088
https://www.walmart.com/ip/Staples-Ayalon-Mesh-and-Fabric-Task-Chair-Black-51505/505776409
https://www.amazon.com/Staples-2722088-Ayalon-Fabric-Chair/dp/B07YBLB56J
Cheaper option: Mellet $220 https://www.amazon.com/COSTWAY-Ergonomic-Armrests-Adjustable-Breathable/dp/B0B17ZRT6N
(I have this chair, and while not perfect, it’s pretty good.)
Mellet is really good in so many ways, but has some weird design decisions. It will fit you pretty well except the arms are pretty far forward and become kinda pointless with the back at maximum recline. The lumbar pad is also utterly obnoxious so I’ve removed it (trivial to do). The headrest is excellent but you’ll need to set it to its lowest position. Apart from those concerns, you’ll get a lot of mileage out of it. The seat pad and back are very cushy and it’s a nice chair to either work or relax in. I purchased this particular one: https://www.amazon.com/COSTWAY-Ergonomic-Armrests-Adjustable-Breathable/dp/B0B17ZRT6N?ref_=ast_sto_dp
With such a good deal on the Ticova and Amazon’s return policies, you might want to buy that and test it for a week. If you like it then you’re done. If not, you can return it and try the Mellet.