Please Help! Is it possible to have two other resonance structures with the double bond being connected to the chlorine.

It is possible for two other resonance structures to be connected but results in positive formal charge which isn't suited for a very electronegative atom such as chlorine.
This is a type of diagram which depicts the bonding between atoms and lone pairs which are present in the molecule.
Adding two other resonance structures will also result in the poor overlap between the p-orbitals.
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