No. You don't need to pay the full price when you upgrade to a new plan. As long as your current plan has not expired, the unused days will be deducted as part of the upgrade amount, which means that you only need to pay for the difference to upgrade your plan.
The pricing/check out page shows you the full price but will only charge you for the difference, please rest assured.
The difference will be credited to your account and deducted from the next invoice(s). If you upgrade your plan, you'll see the credit on the billing page at https://www.flexclip.com/profile.html.