As a start, I am not sure I care too much about consistency of placement (ideally, it would be uniform everywhere, but if this is not workable, I just want to not have obscured New Resource tab).....really just want the Resource page to have the appropriate box show up....any suggestions for template or style properties to make this happen? (Obviously, I don't want the same problem to occur on other places on my site....I am open to suggestions on a "global solution" or just how to fix/change this on the Resource page).You have added margin-top: 28px !important; to #QuickSearch.
You can change that to move it up or down but it will also change it on other pages.
Do you want the search box to be in the same place on every page?
If not, you can add another line of code for .resource_index #QuickSearch specifiying a different margin.