If your drone operations comply with the new Part 107 rule, then there is no need for you to retain your COA, whether it be Section 333 COA or a public COA. However, if your drone operations do not comply with the Part 107 rule (maybe you have not yet obtained a remote pilot certificate which is required to operate under the Part 107 rule) then you should continue to fly under those COA requirements until the COA expires. The FAA has indicated that its policy will be to not renew current Section 333 exemptions if the drone operation permitted under the exemption can be completed under the new Part 107 rule.
COAs for public aircraft operation provide significant latitude to the public entity. For this reason, it may be more advantageous for a public entity to continue operation under a COA. This will need to be evaluated on a case-by-case basis.