Magento – Prices are double at checkout

You don't need to save price for product.

Just pass your dynamic price with in post parameter and set your dynamic price for product in observer event.

You can use an observer class to listen to checkout_cart_product_add_after, and use a product’s “Super Mode” to set custom prices against the quote item.

In your /app/code/local/{namespace}/{yourmodule}/etc/config.xml:

<config>
    ...
    <frontend>
        ...
        <events>
            <checkout_cart_product_add_after>
                <observers>
                    <unique_event_name>
                        <class>{{modulename}}/observer</class>
                        <method>modifyPrice</method>
                    </unique_event_name>
                </observers>
            </checkout_cart_product_add_after>
        </events>
        ...
    </frontend>
    ...
</config>

And then create an Observer class at /app/code/local/{namespace}/{yourmodule}/Model/Observer.php

 class <namespace>_<modulename>_Model_Observer
{
    public function modifyPrice(Varien_Event_Observer $obs)
    {
        // Get the quote item
        $item = $obs->getQuoteItem();
        // Ensure we have the parent item, if it has one
        $item = ( $item->getParentItem() ? $item->getParentItem() : $item );
        // Load the custom price
        $price = "your custom price logic";
        // Set the custom price
        $item->setCustomPrice($price);
        $item->setOriginalCustomPrice($price);
        // Enable super mode on the product.
        $item->getProduct()->setIsSuperMode(true);
    }



}

This is actually a feature.
On the backend the admin can do whatever he/she wants with the product prices. This is normal and a good feature.
But when adding a product to the frontend cart (from backend or frontend) the prices and rules for that website/store view are applied.
There is no way, out of the box to add a product to the customer cart using a custom price.