Leaks after it dispenses water

Maybe and maybe not.
Page 9 of the user manual mentions a anti sweat heater that you can turn on or off.
Samsung SRS579NP User Manual Ice cold Water Dispenser

It maybe that the water line to the water dispenser is leaking
I am including a link to youtube videos about refrigerator water leaks but there is little about Samsung. Hopefully turning on the heater will solve your problem. At least one of the videos shows how to replace the tubing in a different brand of refrigerator.
samsung side by side water leak from freezer door YouTube

Since this is a brand new unit, I wouldn't waste the time or money troubleshooting it yourself; contact the place you purchased it from and exchange it.

check all tubing that leads to water despinser may be cracked or have came lose in chilling coil inside of fidg.

This can be pulled out of the door and replaced with new tubing.You may need to remove the front dispenser panel in order to get the tubing out of the dispenser.

Once you do that then you can pull the remove the tubing and replace it with new line.

Good day,
If the water dispenses properly, then when de-energized continues to dribble, or something like that,
it would be the water valve in the rear bottom of the machine. When you press and ask for water, you are actually depressing a microswitch, which in turn powers the valve. The valve is leaking when unpowered.
You will need to install a new valve assembly, available at any appliance parts house.

Thank you for using FixYa

check the fiting where you tapped the water for the uit. If its a saddle valve what happens is the the gasket closes down and restricts the water flow.

Hello
Welcome to fixya.
I'm Huuum and happy to assist you,

You said, When the water dispenser is pushed you can hear it try to work (a humming or pumping sound) but nothing comes out.
Below is a list of reasons that the water may not be working that are easily checked and remedyed by you! Is there a water shut off on your fridge?If so the dispensor will still make noise as if it is trying, even if it is locked!!!!!!!!!!!!!!!!!!!!!!!

1. When did you change the internal water filter last?
You can remove the water filter and still use the water dispenser to check if the filter may be the problem!If nothing changes reinsert the water filter.

2.Flushing the water system after replacing filter
Use a sturdy container to depress the water dispenser bar until
the water begins to flow. Flush the water system by dispensing
and discarding 2 to 3 gal. (8 to 12 L) of water. Cleaning the
system will take approximately 6 to 7 minutes and will help clear
air from the line. Additional flushing may be required in some
households.
NOTE: As air is cleared from the system, water may spurt out of
the dispenser.

3.***Read this:
The dispenser lock (on some models)
Your dispenser can be turned off for easy cleaning or to prevent
accidental dispensing by small children and pets. To turn off the
dispenser, press the LOCK button. The indicator will appear
above the LOCK button and the ice and water dispensers will not
work. To unlock the dispenser, press the UNLOCK button until
the indicator appears. Then continue to dispense ice or water as
usual.
NOTE:********** The lock feature does not shut off power to the product,
to the ice maker, or to the dispenser light. It simply deactivates
the dispenser levers.

4.The water dispenser will not operate properly

Has the water system not filled? The water system needs
to be filled the first time it is used. Use a sturdy container to
depress the water dispenser bar until the water begins to
flow. Flush the water system by dispensing and discarding
2 to 3 gal. (8 to 12 L) of water. Cleaning the system will take
approximately 6 to 7 minutes and will help clear air from the
line. Additional flushing may be required in some households.
See the “Water and Ice Dispensers” section.

Is the water shutoff valve not turned on or the water line
connected at the source? Make sure the water shutoff valve
is turned on and the water source is connected to the
refrigerator. See the “Water Supply Connection” section.

Is the water shutoff valve clogged or incorrectly installed?
See the “Water Supply Connection” section. If clogging or
installation is not a problem, call for service or contact a
plumber.

Is there a kink in the home water source line? If you find or
suspect a kink in the water line, call for service.

Is there a water filter installed on the refrigerator? The
filter may be clogged or incorrectly installed. First, check the
filter installation instructions to ensure that the filter was
installed correctly and is not clogged. If installation or
clogging is not a problem, call for service.

Please remember to leave a rating before you go!
Thank you,
Huuum

afraid that sounds exactly like the problem.brand new fridge call who you bought it from they should fix it for free good luck hope this solves your problem

If it is a plastic compression fitting you can tighten each one a little(careful not to tighten them too much cause they can strip out) Use a small crescent wrench with a back up wrench.

It sounds like you have a clog. If the clog is at the nozzle, you would still have pressure, however nothing would come out. It could also be the valve that opens for the dispenser.

Take it apart and look. I would focus on the dispenser end.

Good luck!