If you are talking about OOTB the path of least resistance would be to bulk unschedule then bulk reschedule, although that wouldn't allow you to select a specific day to move them to.

If you wanted to get that granular you could incorporate a Flow, VF Page, and a Custom Gantt Action. Select all the SAs, kick off the gantt action (either from the list view or right clicking one of the SAs), your flow would allow you to select a date, then the flow would go and update the dates accordingly. You will want to be mindful of the earliest start and due date to ensure that you don't attempt to move outside of those boundaries.

Using something like optimization could definitely be a potential help here, although it depends on the workflow that is being followed.

I would have questions on the use case overall though. I understand that there is just manual drag and drop currently, but even with drag and drop, a dispatcher can see that a technician is getting full with appointments, right? So I would just be curious what the catalyst is to determine that the tech now has too many appointments, more than they will be able to handle, and now some of them need to be moved?

I would also want to investigate the impact of this type of workflow on overall customer experience (if appointments are booked with Customers) or overall work completion since it seems work is continuously pushed out on a day to day basis.

I would really want to try to work backwards and understand how we can fix the problem of having to move work out, which could be investigation into using the other scheduling tools within Field Service including optimization.

Keep in mind I have little to no details on the nuances of the use cases you are solving for, so please take any feedback as such. More just observations and thoughts based on the details provided.