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???
If you can wait that long. It'll probably confirm within 3-7 days. If it doesn't you should try making another payment from the same address with a higher fee.
You can use accelerators like: pool.viabtc.com/tools/txaccelerator to accelerate your transaction. Or search CPFP and RBF double spends and see if you would be able to do one of those.
Also, these don't need the same wallet to be used, they need the same private key to be signed - there's a difference.