Trying to raise scores

The most logical choice in an attempt to raise your scores would be to dispute the old (and now paid off) derogatory accounts. Although they are paid, they are likely still depressing your scores insofar as a derogatory is a detriment regardless of whether or not it is paid.

You'll want to examine the account tradeline "status" as well as the dates of initial delinquency. Reviewing the dates of status and balance histories is a good idea as well inasmuch as they should match up. These are a few of the data fields that you may elect to dispute. Moreover, if any of the account tradelines are furnished by collection agencies or debt purchasers, you may as well go ahead and dispute the validity of the underlying debt with them directly. It is irrelevant that it is now paid rather, what you're driving at is to see if they will mark the account as "in dispute." If they fail to do so, you have an actionable FDCPA (and FCRA) violation which you can leverage to your advantage in terms of deletion. Of course, always "dispute" tradelines through the credit reporting agencies so that you can obtain actionable violations against said furnishers.

With that said, it is likely that even if you remove all of the derogatory items, your scores may only be marginal. The reason for this is that the key to a healthy credit score is little to no bad items and a voluminous amount of good items (or even a smalll amount with necessary history). In other words, you will need to add some positive tradelines (preferrably with some age) UNLESS you already have this requisite.

There are numerous ways to accomplish adding tradelines. First, once your scores are around 620, apply for an Amex Delta Sky Points. There is a link for it here. Maybe opt for a Union Plus by HSBC or a Juniper Bank underwritten card. Citibank's Premier Pass and Applied Advantage are also options. However, if you're shooting for a mortgage in the near future, this is not the route to travel. You will need to add some aged tradelines insofar as opening new accounts will typically lower your scores. Conversely, adding some aged accounts generally increases them with the longer the account history adding a greater number of points. I would suggest asking a friend or relative to add you to some of their accounts IF they are and have always been paid as agreed and IF the utilization is below 30% of the available limit. If this is not a viable option, there are companies which offer such services.

Finally, and FYI, you can obtain a mortgage now. The rates may be around 8% or so at 100% financing on a 30yr fixed but it can be done. You just need to browse around.