Good morning Aquintanilla,
You are being faced with a question i've seen for years as not only a Certified Credit Counselor through NFCCC, as an approved trainer of the Freddie Mac curriculum "Credit Smart", financial director of education with CCC Centers, but also a Realtor that personally instructs and guides my clients with their lender's advice.
The crux of the whole thing is even Fair Isaac Corporation (FICO) does not release their formulas as to how scores are calculated. Everyone is simply doing as much homework as they can and trying to "break the code". And some are getting very close, for example.. it now can be proven that by having a Capital One credit card can actually hurt your credit even if it's current and never over the limit.
By past due accounts you are referring to revolving consumer debt (credit cards) or installment debt (car payments), then almost always the right thing to do would be to bring those past due account current and/or pay them off (again... even that can be tricky.. it's a fun credit game).
Or by "past due accounts" you are referring to old collections (any account turned over to an outside company for collections), then the answer is not as clear.
This is where it really helps to have a mortgage lender (one that really knows what they are doing and can understand credit) and seek advice from them. Typically, they will consult with their underwriters on what exactly they would like for you to do. It's truly your best answer. Remember, you are only concerned about your credit so you can obtain more of it (ie Mortgage), therefore consult w/ the actual lender you are looking to obtain a mortgage through. It's much better to get it "straight from the horse's mouth" then try and stab away in the dark by listening to everyone's opinions on what you should do. Unless they are willing to step up and give you the loan (I'm even talking about the credit repair company here), then I would seek counsel from the source.
Best of luck and God bless you on your venture