All judgments entered against you will be picked up by the credit reporting agencies. Those who review the public records will most likely not pick up on the factor whether the judgment is "in rem" only, meaning you are being sued only with relation to the property securing the loan, or "in personam" which means you and the property.
But, here is the reality, your credit score is going to be severely adversely affected by the filing of the bankruptcy, so I really do not know what you think you will be saving by avoiding having judgments entered against you. You will not to rehabilitate your credit before asking a potential creditor to review it.
With that having been said, let me ask you to consider the wisdom of keeping properties with negative equity. I understand that you may be generating positive cash flow if the properties are rented out, and you are correct that the Trustee is extremely unlikely to take the property, but what is the overall end game?
For example, let us take the situation wherein you keep the rental properties, life goes along, you rehabilitate your credit, things are better.
Then, for whatever reason, whether by your loss of a job, or illness, or the tenants not paying, or a hurricane hitting and only partially damaging the structure which means tenants would have to leave, or if the properties aren't being rented out and you just plain old lose the ability to make the payments on the mortgage.
New foreclosure actions--and new judgments are entered--will deliver more blows to your credit ratings, so that the time you spent after receiving your bankruptcy discharge will go for naught, as new negative notations will be hitting the credit agencies. Remember that the bankruptcy discharge does not wipe out the debt, it wipes out your obligation to pay the debt.
I do not mean to lecture, and there is an awful lot that is not revealed in your question, but I believe the strategy of keeping the properties that are in negative equity may cause you grief later on in life.
Bankruptycy is not my speciality. I do not like Bankruptcy, but somtimes it is the only choice. What I can tell you is that if you have an attorney, you should ask them. If you are planning on getting an attorney get one as soon as possible to file bankruptcy. If you are doing it yourself, do it ASAP. It sounds as if you are trying to time the bankruptcy by picking the "right time". If I was your creditor's attorney and was after your money, I would try to argue that you "should have filed sooner and that you waited too long." If you know the ship is sinking get on a lifeboat as soon as possible.