One airdrop has been asking me to cash out the tokens using my TON wallet, but when I try to, the prompt I receive is that I do not have TON to complete the transaction. Must I have TON to receive Telegram-based airdrops?
Yes, it is a condition but you can use My Ton wallet or Tonke3eper extension ( Link is provided below). The airdrop offered on telegram channel are becoming very popular because they require minimal investment to participate while promising rewards are expected. With active users on telegram more than 700 Million and this massive community is its significant strength and important factor for launching projects on it and attracting investment.
https://play.google.com/store/apps/details?id=com.ton_keeperYou are right. Sometimes it's for free (as it was with some memecoin I've participated in, $ALT), and sometimes it's for a fee, however, it really shouldn't be a lot of money to spend to get the drop on your wallet.