My dictionary shows 说 is both shuì and shuō I have installed Chinese character dictionary popup plugin into the browser and it showed for 说 character:

说   說   shuì   to persuade
说   說   shuō   to speak; to say; to explain; to scold; to tell off; a theory (typically the last character in a compound, as in 日心說|日心说 heliocentric theory)


Is this correct? Why does shuì go first?
Yes, it's correct. 
shui4 说客 说士 lobbyist 说服 persuade 说导 enlighten etc.
However considering that many people today mix them up In pronunciation, 说 is supposed to be pronouned as shuo1 In both meanings, according to the latest edition of 《现代汉语词典》, with many other pronunciation adjustments as well.
The whole thing is controversial. Highly educated people refuse to succumb to the mispronunciation, though so prevalent nowadays.