This is The Enchanted Forest Chronicles, by Patricia C. Wrede. It's a 4 book series. The books are Dealing with Dragons, Searching for Dragons, Calling on Dragons, and Talking to Dragons.
The first book was originally published in the UK under the title Dragonsbane, so you may remember that title instead.
One cover of the first book (you may not have seen this particular cover design, there are others)
Blurbs from wikipedia, for the first book, Dealing with Dragons:
In which Princess Cimorene of the kingdom of Linderwall decides that being a princess is too boring and confining, leaves home to work for the dragon Kazul, and discovers and subsequently dissolves a plot by the wizards to take control of the King of the Dragons.
Searching for Dragons:
In which Cimorene meets the King of the Enchanted Forest, Mendanbar. With their new friends, they collaborate to rescue Kazul—now the King of the Dragons—from the wizards who have captured her in the Enchanted Forest.
Calling on Dragons:
In which Morwen discovers that the wizards have stolen Mendanbar's sword, which kept them from stealing the Enchanted Forest's magic, and works with Cimorene to retrieve it.
Talking to Dragons:
In which Daystar, Cimorene and Mendanbar's son, is sent off into the forest with his father's sword and no knowledge of his heritage. It was written and published first, and then revised later to better fit with the prequel books.
Your descriptions, in the order you gave:
There's a magical sword belonging to the king of the forest that evil wizards want. The sword isn't only in this book, but Daystar is. Daystar is the son of Cimorene and the only person who can handle the sword,
as he is the heir to the position of king of the forest. He's Mendanbar's son.
In the third book, Calling on Dragons, they end up trekking through a swamp full of invisible plants that will strangle you.
A major plot point is:
the king of the forest, Mendanbar, is trapped in the castle by the wizards, and only his heir (Daystar, his son) can free him.
The wizards melt in soapy water with a hint of lemon. The good guys eventually make a spell to squirt this on the wizards, so they don't have to lug buckets around.
Definitely accurate, almost no people die. A dragon is poisoned in the first book, but that's all I can remember.
Cimorene runs away from home in the first book, because she doesn't want to be a traditional princess. She ends up being a "princess" to the dragon, basically a maid.
The dragons were mostly good, though some weren't very nice. In Dealing with Dragons,
a dragon named Woraug poisons another dragon to try to usurp the throne.