Magento – How to Show Custom Product Price on Listing Page

This could be the problem:

<img src="../images/sale-icon.png" width="101" height="58" class="onsaleicon" />

When you are not on homepage the relative path to the image is wrong.
Use this instead:

<img src="<?php echo $this->getSkinUrl('images/sale-icon.png')?>" width="101" height="58" class="onsaleicon" />

and place your image in skin/frontend/{interface}/{theme}/images/.

Small tip:
Don't use this:

$specialprice = Mage::getModel('catalog/product')->load($_product->getId())->getSpecialPrice(); 
    // Get the Special Price FROM date
    $specialPriceFromDate = Mage::getModel('catalog/product')->load($_product->getId())->getSpecialFromDate();
    // Get the Special Price TO date
    $specialPriceToDate = Mage::getModel('catalog/product')->load($_product->getId())->getSpecialToDate();

You are loading the product 3 times and you don't need to do that. You can load it only once if neccesary:

$mainProduct = Mage::getModel('catalog/product')->load($_product->getId());
$specialprice = $mainProduct->getSpecialPrice(); 
// Get the Special Price FROM date
$specialPriceFromDate = $mainProduct->getSpecialFromDate();
// Get the Special Price TO date
$specialPriceToDate = $mainProduct->getSpecialToDate();

but you can go without loading it, by editing the attributes 'special_price', 'special_from_date' andspecial_to_date` and settingused in product listingtoyes`.

bigger tip:

You can check if the product has a discount without checking the special price and the dates. This works even if you have a discount from a catalog rule:

if ($_product->getPrice() > $_product->getFinalPrice())

Using the handles: catalog_category_default, catalog_category_layered and catalogsearch_result_index you should be able to change the template of the products list.

For the two category layout handles you would need the following snippet.

<catalog_category_default>
    <reference name="product_list">
        <action method="setTemplate">
            <template>MyModule/Availability/catalog/product/list.phtml</template>
        </action>
    </reference>
</catalog_category_default>
<catalog_category_layered>
    <reference name="product_list">
        <action method="setTemplate">
            <template>MyModule/Availability/catalog/product/list.phtml</template>
        </action>
    </reference>
</catalog_category_layered>

But for the search results page you would need to make a slight change as the block is not called product_list but search_result_list:

<catalogsearch_result_index>
    <reference name="search_result_list">
        <action method="setTemplate">
            <template>MyModule/Availability/catalog/product/list.phtml</template>
        </action>
    </reference>
</catalogsearch_result_index>