how to send jettons with TonWeb? or is there any other way to do it programmatically?

You can check https://github.com/toncenter/tonweb/blob/master/src/test-jetton.js, I hope it will be helpful for you

I just checked it and tried to do the same thing, but getting some js errors saying that it can't parse output from provider or something

Are you sure with JETTON_WALLET_ADDRESS is correct for your case? It's should be jetton-wallet-address (another contract) not classic wallet (v3/v4) address

Better to see logs/output/code. If you want, can chat me in direct with logs

Oh, that was the problem! Thanks, I forgot that they're different contracts