This won’t solve the problem and they aren’t a necessity but, once you’ve achieved appropriately cooked rice, will slightly improve the seasoning process:
It’s a sushi oke or hangiri, and whilst it is better to buy them in Japan where you can give the wood a sniff first (the classic kind is made of cypress) so long as the wood has some absorbency you should still get a better texture combining hot rice with vinegar in one of these. The one in the link is likely to be pine wood.
The price for these - which ideally come with a lid - is very affordable in Japan
https://shimizumokuzai.jp/en/sushioke
This link has excellent instructions on how to wash and get a sushi oke ready before using it for the first time.
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Another way to improve vinegared rice is to carry out that extra suggestion in the rice making instructions at Step 5, Resting. The easiest way to do it is to have a second lid handy with a tea towel secured onto it that covers the underneath part of the lid. Once the rice has finished cooking, very, very quickly, losing as little steam as possible, swap over the lids.
Without a tea towel, or paper towel, lining the top, once cooked, the steam inside the pot is going to collect onto the top, condense and then drop back onto the rice. The cooled drip water won’t be reabsorbed consistently and will contribute to making rice feel wet.