Well, my suggestion would be to take some tin snips or gardening pruners or whatever, cutting each of them once to break the circle, and dropping them into moving water like a river or creek.
Cutting them so the circle is broken would symbolize the broken engagement, and dropping them into moving water would wisk that broken engagement away from you, not making you forget, but taking it away from you and out into the world, releasing you to your new relationship. Sort of like a cleansing ritual for your mind.
The way I see it, engagement rings, wedding bands, promise rings, etc, hold emotions and memories. By keeping them in any form, melted or not, you are holding on to the emotion and memory of your ex. It's better to give those to nature and let it heal you than to hold onto them. I've also never been a fan of giving them away because I really do think that some items like these actually hold that emotional charge and could affect the next person that gets them.
When I got divorced, I did this with my wedding band. The gold was only worth about $3 at the time so no money lost there. My dd wanted her dad's wedding band, and she kept it for about 6 months before she had to get rid of it because of the emotions she felt with it. She gave it away, and almost immediately got it back because the person said it made them feel oogy. She asked her dad if he wanted it, he said no throw it away. So I had her cut it and toss it into a creek. Right after that is when she finally started feeling better about herself and our divorce, and started treating my now DH much better.