I have a couple of things to point out, the tasks priorities are not responsibility of the development team at all, this is something that is managed by the PO for many reasons, but lets not discuss about that.
As you don't have a single PO this is transformed in a fight of interests. If there is no common goal between them every PO will want their US to be done ASAP because is the highest priority for them (and this is totally understandable).
So, if you want to keep one single team for all this products you will need a PO that works as single point of contact for the Dev team, a single backlog for them and then you can try to make sprint goals focused on single products so the dev team does not have to change their "environment" in the middle of the sprint (but this is a bonus point, not mandatory).
The main thing is if it is possible to have this single PO to manage this single backlog, at the end its the same as if your current 5 POs becomes stakeholders, they will ask for what they want, and this single PO will put those thing in order. Based on what ? could be company need, if there are blocking issues or it could go as easy as money ... who is the one that is paying the most and thats the one you will attend first.
In resume, remove the responsibility of picking the task to be developed off the team, it could be by this single PO approach, by a forum between the PO where they manage the single product backlog all together. And those are my 2 propositions.
There are too many factors in place, the company, the POs shared vision and needs, the reason why there is 1 single team that is managing all this.. etc.
We are currently working with one single team for multiple products as well and we have just one PO and a single product backlog and things goes smoothly.
Hope this helps! Any doubt just tell me!