Examples can be found in Scripture(Deut. 19:15 and other passages)
I like the idea of having a priest cancel a vow made in public.
Yep, it is in scripture you are right. It seems you are refering to public vows, like to tell the truth in court, or offered to the church: vows of a nun, marriage, priest, etc
It appears I am right about the other type of vow; private vows (which a layperson can make in their mind on their own)
Here are some quotes from the catholic catechism and catholic canon law;
The Church Canon Laws, in certain cases and for proportionate reasons, dispense from vows and promises [Cf. CIC, cann. 692; 1196-1197]"
Q. 2. The above references appears to refer to those who make religious vows. What about a lay person who made a private vow? What if I made a private vow and I no longer wish to uphold that vow? What do I do?
A. 2. Many, with all good intent, when they are young and zealous, they make a vow to God with the intention of keeping it for life. Not having consulted a spiritual director prior to making such a vow, they do not realize that a vow for life is a long time. Also, those who make private vows often walk a lonely spiritual life without the support that religious vocations find in Dioceses or religious Orders.
Such vows or promises may consist of a vow to remain celibate for life, a promise to be a missionary for life, a promise to pray the Rosary on a daily basis throughout one’s life, a promise to never commit a certain sin, etc…
Knowing that many individual do make private vows and that some may want to be dispensed from such a vow during their lives, the Church covers this matter in the Canon Laws where it states:
Canon Law # 692 “An indult to leave the institute, which is lawfully granted and notified to the member, by virtue of the law itself carries with it, unless it has been rejected by the member in the act of notification, a dispensation from the vows and from all obligations arising from profession.”"
Canon Law # 1196 “Besides the Roman Pontiff, the following can dispense from private vows, provided the dispensation does not injure the acquired rights of others;”
1° “the local Ordinary and the parish priest, in respect of all their own subjects and also of peregrini;”
(Note: “peregrini” means "travellers.)
2° "the Superior of a religious institute or of a society of apostolic life, if these are clerical and of pontifical right, in respect of members, novices and those who reside day and night in a house of the institute or society;
3° “those to whom the faculty of dispensing has been delegated by the Apostolic See or by the local Ordinary.”
Canon Law # 1197 “What has been promised by private vow can be commuted into something better or equally good by the person who made the vow. It can be commuted into something less good by one who has authority to dispense in accordance with Canon Law # 1196.”
(In simple English, # 1197 means "The person who makes a private vow can commute the work promised by the vow into a better or equal good; however, one who has the power of dispensing according to the norm of ? can. 1196 can commute it into a lesser good.)
To answer your question, private vows that are not associated with a Diocese or a religious Order, falls under the juridisction of the parish priest (see # 1196. 1° above). In order to have your private vow or promise replaced by something else of equal good or of a lesser good, please consult your local priest.
Lives of the saints show personal private vows are binding too;
Original poster made a resolution -which is not binding, and not a vow.