There seems to be a difference between a change of mind, and being interactive, and a change of mood.
Earlier God gave them the choice:
Deuterenomy 30:15-18 (NIV)
15 See, I set before you today life and prosperity, death and destruction. 16 For I command you today to love the Lord your God, to walk in obedience to him, and to keep his commands, decrees and laws; then you will live and increase, and the Lord your God will bless you in the land you are entering to possess. 17 But if your heart turns away and you are not obedient, and if you are drawn away to bow down to other gods and worship them, 18 I declare to you this day that you will certainly be destroyed. You will not live long in the land you are crossing the Jordan to enter and possess.
The situation in Judges refers to the fact that the nation did not continue to follow him, so they had to face the consequences as stated before. They were about to be destroyed, and God did not intend to help them, if they would not worship him and worship other gods (as in the agreement in Deuteronomy). And God informed them about it In Judges 10:13, as you quoted.
In response to his warning for the coming destruction, they responded (Judges 10:15):
15 But the Israelites said to the Lord, “We have sinned. Do with us
whatever you think best, but please rescue us now.” 16 Then they got
rid of the foreign gods among them and served the Lord. And he could
bear Israel’s misery no longer.
So because they repented and changed their behaviour, God reacted according to their reaction and saved them.
Since the messages and actions of God in these examples are all according the earlier stated principles, it would be harsh to say that God was self contradictory or changed his mind. He just reacted according to their behaviour, and how they used their free will.
To me the example closest to change of mind in the bible is in Genesis 6:5-8 (NIV)
5 The Lord saw how great the wickedness of the human race had become
on the earth, and that every inclination of the thoughts of the human
heart was only evil all the time. 6 The Lord regretted that he had
made human beings on the earth, and his heart was deeply troubled. 7 So
the Lord said, “I will wipe from the face of the earth the human race
I have created—and with them the animals, the birds and the creatures
that move along the ground—for I regret that I have made them.” 8 But
Noah found favor in the eyes of the Lord.
It seems that Gods state is adaptive to human behaviour and free will. Assuming divine inspiration, it is interesting that Gods promises to stand by his principles, but also wants to express the "feelings" (that it is difficult to do so), when people are abandoning his standards all the time.