I think there have been several discussions on the boards about this.
But here are a few of my thoughts:
If you only have on polished PB manuscript, it's too soon to be looking for an agent. If an agent likes your manuscript, most likely they will ask to see more from you. Since PBs make very little $ for agents, they want to feel confident that they could sell more of your manuscripts.
So, if you only have one PB done, you could take a few more months and write at least one or two more and get them as polished as possible. When you start querying, you would only submit with one, but if they ask for more, then you are ready.
OR you could start querying publishers who take unsolicited manuscripts. As you noted, there aren't too many of those so your options are limited. But there are authors who have created successful careers without agents.
One other thing to consider: many writers decide to start with agents first. If they aren't successful then they try the editors at houses that accept unsolicited submissions. It doesn't make sense to do editors first, then agents, as if you would find an agent who likes your work, you would need to be honest about where you have sent the manuscript. If it's more than a handful, then the agent might consider your manuscript "shopped around" and have less places they can submit and that may lead to the decision not to represent you.
I hope this helps.