The only way to deal with the change is to pump, bail, or drain the water released as you dig out blocks. Bailing takes far too much time, and draining is hard to control. Pumps are the best option but you need to beat the goblin invasion for it. Before I have the Tinker & Mechanic, I dig around water, not through. If you want to place an air pocket underwater, you need 6 blocks and at least one empty bucket.
It's always easier to deal with the reduced amount of water you leak by using the old tunnel method than trying to drain the entire body of water.
Note that when a block has been placed in water and is "wet", the hookshot and ivy whip tend to release from the block immediately after attachment. For this reason, I like to build vertical tunnels through water one block wider so I can place dry walls after I pump out the water.
ASCII diagram: (X is dry bricks, O is empty space, W is water, Y is wet bricks)
XOOX XOOX XOOX XOOX
XOOX XOOX XOOX XOOX
XOOX XOOX XOOX XOOX
XOOOOX ---> XOOOOX ---> XXOOXX ---> XOOX
WYWWWWYW WYOOOOYW WYXOOXYW WXOOXW
WYWWWWYW WYOOOOYW WYXOOXYW WXOOXW
I build a 3-block wide shaft, but the principles are the same.
When building a horizontal tunnel that you expect water to drain through, put torches either in ceiling notches or underneath glass floor (only one glass block directly above the torch is needed, but I like to use 3 for aesthetics).
I strongly recommend you prioritize getting a goblin invasion asap so you can rescue the two new NPCs and build pumps, pumps are fantastic.
It might still be possible to destroy water with falling sand or silt, but I haven't tested it because I had pumps when I started digging my hellevator. That would provide an early-game alternative to pumping away the water you leak by tunneling. If you have tested, feel free comment or edit this answer. Falling sand no longer destroys water.