Don't use regular toothpaste because it's too abraisive and will scratch the denture which in turn will give bacteria a good home to grow in
. You don't want that
As for soaking overnight in a cleaning solution or plain water, the partial won't melt or otherwise self destruct. As a matter of fact, when dentures are made they are heat cured so hot foods or even baking them into a pie won't melt them
One of the better cleaners for dentures is probably in your bathroom already; hand soap
. I soak my denture overnight in cleaning solution and brush them with soap each day. They're as clean and stain free as when I got them 14 months ago.