Jayeff has given you some great recommendations.
In relation to your specific question "Is my phone beyond repair or can it be saved?", the answer is yes, it can probably be repaired. The phone works as is for the most part. Water probably leaked in via the speaker mesh (hence no speaker functionality) and the SIM card tray.
The problem with water damage is its insidious nature. You can pay to swap out the screen, the speaker and maybe have the SIM card reader replaced. However, there is always the danger that more damage is occurring underneath the electronic chips on the board, most of them tucked away by shields.
Your phone needs to be professionally decontaminated to stop the (hidden) corrosion for progressing. Then you can look at how much it will cost to repair and determine if its worthwhile compared to an OOW replacement with Apple.
Once your device is wet or repaired by a third party, your warranty is already over so don't give that another thought. The iPhone 5S/SE is a great phone and well designed. It has very few known flaws (unlike the 6/6+) so its certainly worth the investment.