So running the full node will help you with the validation process quickly and broadcast the transaction to mempool which means they will be in the unconfirmed status.
You do not need to run a full node to be able to broadcast a transaction to the mempool, any wallet can do that and other nodes can validate the transaction after which they add it to their mempool.
They are incentivized to include a transaction with higher gas fee to earn those as reward for mining a block.
It is not called gas fees in Bitcoin, but it is called transaction fees.
- Jay -