What should I pick for sportswear if synthetics are bad (microplastics)? It's not a pleasant experience to sweat in cotton clothes (whether organic or not).
Use a fine-mesh bag when washing your sportswear
The first goal of sustainability is to reuse goods for as long as possible. Microplastics are a real concern, but we don't need to throw our synthetic textiles into the trash. Instead, make sure to put your synthetics in a fine-mesh bag (like the Guppyfriend*) when doing your washing. This will trap at least 90% of the fibres that break off, substantially decreasing the amount of microplastics that end up in the ocean.
*I have no affiliation or connection with this brand, I just know of them as a good solution to this problem.
Consider natural fibres
Natural fibres like wool (from animals) and hemp (from plants) can be woven to produce fabrics and textiles with desirable properties for sportswear. Merino wool is moisture-wicking and retains its insulating properties even when damp, making it a good choice for activewear. Vegans may prefer to avoid animal-based products, so they should look toward hemp-based apparel as one of the most sustainable fibres for producing textiles.
The sustainability of synthetic fibers cannot be reduced to microplastic shedding. In general, polyester (and other synthetic fibers) use much less water during production, no pesticides and have other environmentally positive aspects like their recyclability (at least to some extent) as well as longer life cycles. So while swapping polyester sports clothing for wollen or cotton ones might reduce microplastics, the swap might bring about other detrimental effects.
Buying as little clothin as possible with the highest possible quality (that lasts the longest), washing less often and on a gentle cycle and environmentally friendly detergent (while still making sure that the clothing gets sufficiently cleanded, increasing its usability duration) and repairing existing clothing is the way to go for environmentally concious clothing consumption.
A couple of further suggestions, that don't involve buying new products (though a similar bag could be made from an old pillowcase):
Use a gentle wash cycle, to reduce the agitation that releases microplastics. I've seen conflicting advice on the effects of soaking and whether we should prefer a cycle that uses a lot of water or a little.
Save the synthetics for when they're needed: If you're outdoors all day in all weathers, carrying all your kit, choose stuff that's lightweight, wicking, quick-drying, etc. If you're just going to the gym, or for a short run in nice weather, a cotton T-shirt might not be so bad after all - probably not your best ones as they can end up less than fresh even after a good wash if you have to save up your washing to make a load. If you exercise frequently, this can reduce the amount you need to own, and prolong the life of synthetic garments.
Another thing you can do to reduce the amount of microplastic synthetic garments shed is to make then last longer. This might mean repairing them, such as replacing a failed zip as I did on a bike jersey yesterday. For a few minutes and a few grams of (synthetic) materials, I significantly prolonged the life of the garment.
I've been researching a similar topic on how to avoid synthetics all together, including outdoor rain gear, hiking, camping, and all that.
Two points, first, before you decide microplastics are bad, might be worth considering this question that I've posted (waiting for answers); the research on the harmfulness of microplastics is very limited at the moment, it may be too early to conclude its effects.
Second point, 100% merino wool is a commonly used as a base layer for hikers; it is soft, dries fast, wicks moisture, and regulates temperature. Otherwise, cotton is not so bad as long as you have something dry to change into. In hot weather, cotton gets wet from sweat and this can also cool you down. The shape of the fabric is also important – how much airflow you get. But you say you don't like feeling wet, so I guess merino is your answer.