I realized I didn’t know how Sparrow Wallet handles broadcasting of transactions, so I looked.

Craig made this announcement on Twitter back in May of 2021 when v1.4.1 was released:

“If Tor is enabled, Sparrow will create a new Tor circuit to broadcast your transactions via @mempool or @Blockstream.” [1]

Reviewing the source code, this is still the case, but there are now more broadcast sources that Sparrow will try. It tries them in random order until the broadcast succeeds. [2]

If you are on MAINNET, it will try the following sources [3]:

blockstream.info

mempool.space

mempool.emzy.de

mempool.bisq.services

So make sure you have a Tor proxy setup and make sure you are connected through that Tor proxy and Sparrow will send your transaction via Tor using one of those public services. This is equivalent to connecting to one of those services using the Tor browser from the device Sparrow Wallet is running on and then using that service to broadcast your transaction directly. But this has the added benefit of using a different randomly chosen service each time you broadcast a transaction.