ANSWER - There is no one answer to this and I think it is important that you are guided by what feels right. The running water thing as you said is a common way of disposing of a poppet but depending on what the poppet is made of this may have implications for the environment. I would only dispose of a poppet in running water if it was properly biodegradable (I sometimes string twigs together in the form of stick men to use as a poppet).
Other good alternatives are burial of the poppet (again this is only an ideal solution if the poppet is made of a substance that will easily break down into the earth) or tying to a tree in the middle of a wood and leaving to decompose in the elements. However if it is a poppet made specifically to depict yourself I would be very reluctant to embrace any method of disposal that leaves the poppet intact in case it fell into the wrong hands. I'd suggest you either need a permanent method of destruction or you retain the poppet in your keeping.
My own preferred method of disposal is to collect all my disused spell workings together every few months or so and then burn them in the back garden. If you do this I would suggest you disassociate yourself from the poppet first - this can easily be achieved by dismantling it and breaking the link with yourself by covering in salt.
There is no reason you cannot keep the poppet and if it feels more right to you then it probably is the best option. However if you are not intending to use it again then it would make sense to perform the disassociation anyway in case it gets lost or falls into the hands of others.