The order of operations
Bind the new policy first, with an effective date. Cancel the old policy effective that same date, by signed request. Confirm the carrier's cancellation notice. Re-issue certificates — or let the live certificate update them.
What it costs
Short-rate cancellation keeps a penalty from the unearned premium; pro-rata returns it all. Policies with minimum earned premiums keep at least that amount. Audited policies bill the final premium for the months in force regardless. Financed premiums are returned to the finance company first.
Doing it from the wallet
Policies serviced by CoverMatch have a cancellation request on the policy page. You e-sign it, it goes to the carrier automatically, live certificates move to "cancellation scheduled" so holders are told in advance, and the carrier's confirmation closes it.