It sounds like you should get clarification from your Loan officer. a couple things concern me about your post.
For Starters, I would bet that you have a closing date without a "Clear to Close." That is the term used by lenders when there are no longer any "conditions" left before they will actually make the loan. If you had a "clear to close," I guarantee your loan officer would move up the closing date.
Secondly, your loan officer needs to be more specific with you about what needs to happen in order for you to get this loan. A collection from 2001, many times, is not something you necessarily need to pay.
If your LO (loan officer) is not being forthright with you, you need to ask some specific questions:
1.) Do you HAVE to pay the collection to get the rate she's quoted?
2.) Why would making a partial payment affect approval or denial? (The short answer is that in most cases, it will not.)
3.) What other options are aavailable to you if you do nothing about the collection?
If your LO does not have a quick and convincing answer to the first two questions, and have some other options for question three, I would seriously consider finding another loan officer. I'm not saying this because I want your business. I don't even deal with consumers anymore. I work for a lender and work with brokers...not consumers. I hope this helps.
Let us know what your loan officer says. Good luck.