So my question is, how do I make my drop-down menu accessible on touch screen devices? Such as the tablets and smart phones
You don't, because you can't. Not directly anyway.
What you can do is have another way to access the same information, such as a sitemap-style link users can click on to reach a page which basically contains all the same links as your dropdown.
You might also want to see if, with CSS media queries, you can only introduce the dropdown CSS on screens above a certain size (though this DOES assume larger screens != touchscreen, which is the case mostly today but may not be in the future... also assumes devices and browsers understand media queries and they may not). If you can get something like that working then the single HTML menu can be an open list on smaller screens and a dropdown on larger screens.... again, this only works on assumption that small screens might touch and large ones never do.
So I like the additional link before or after the menu better. I would agree with ralph about dropdowns not being all that great in the first place, but I've built them before because clients have demanded them, so I know how that is.
Always nice to have some redundancy in stuff like navigation.