Woocommerce Remove Price If Price is Not Set

Based on the answer to How to display woocommerce sale price or regular price if there is no sale price, I have a product loop that displays the product price with a "$" sign before the price. The problem is that products without prices still show "$".

Based on the answers to Using AND/OR in if else PHP statement and wordpress/woocommerce: Remove price if zero, I tried to add an additional if condition using && but can't get it to work:

In functions.php:

function get_regular_or_sale_price() {
    global $product;
    if ( $product->price && $product->is_on_sale() ) {
        return '$'.$product->get_sale_price();
    }
    return '$'.$product->get_regular_price();
}

function get_regular_price_if_sale() {
    global $product;
    if ( $product->price && $product->is_on_sale() ) {        return '$'.$product->get_regular_price();
    }
    return '$'.$product->get_regular_price();
}