If my goal is to get it off the network ASAP, Is it better to stop broadcasting the transaction or keep it going?
If it's better to stop, how can I do that? Simply by not opening my wallet(s) to check every 5 mins??
What wallet do you use? If you use a wallet like Bitcoin Core, then it's best to use an option like -zapwallettxes that drops all pending transactions. Else, your transaction will get re-broadcasted as soon as you open your wallet again.
If you have a wallet that sends coins with RBF option, you could replace the tx by a new tx with higher fee.
Third option is to use a transaction accelerator.
I used blockchain mobile wallet and chose the fee they suggested.
Really, they suggested 4 sat/byte???