I tend to combine burrows with priorites (and with the Marker Only and Marker-toggling, too).
I may have priority 1s in various parts of the digging-to-expand, with 2s and lower as catch-alls. But I also use burrows to make a particular miner (or several of them, as necessary) re-override any other priority tendency.
A good trick I use is to priority-2 a skeleton of linking through a priority-3 room (for example, leading to the opposite wall of a bedroom where I want to place the bed) and then burrow-define the area so that a constrained miner will go for those 2s (clearing out the bed/etc build position(s) first, doorways to other rooms, spot in the middle where the workshop will be, as appropriate to that particular plan) and then start on the 3s to open the place out, just needing my monitoring (at the most) not any further active micromanagement until the final 3s are dug out (or 4s, 5s, 6s, even 7s, depending on the complexity of the pre-planning).
I also do (with either these numbers, or shifted ones to match other plans) a set of rooms as 2s, tracks across the rooms as 1s and doorways into/between the rooms as 3s. The miner(s) concentrate on any 'exposed' rooms, cutting out tracks across them, before emptying the rest of the area, but only as soon as (deprived of access to anything less) the first find an external (or the next internal) doorway 3 and thus force work to concentrate in that area, until it doesn't need to and can start to expose another room entry.
But as I may have bedrooms in one part of the fortress, an area intended to be a granite-item multi-manufacturery in another part, a cistern needing dug in another area, exploratory stairwells looking for the magma elsewhere, the ground-level ditching (first of all the footprints of the bridges that will be laid over them, then the rest of the ditch after that, again with priority levels dictating some sort of order) and probably also a new emergency trade-goods stockpile room beneath the depot, I will create a burrow, each, expressly over every digging-zone that I designate (and also food/drink/sleeping areas, usually, just to keep a confined digger slightly more content than otherwise) and then when I think "I really want that cistern finished ASAP" or "I still need to find that magma" I assign one (usually, and choosing one that I may know has just come back to the rock-face eating, to maximise their time off-break) of my miners to the relevent burrow, and that element gets priority (dedicated mining by that one individual, that is). Even while another of my miners might be suitably prioritising a different digging-project. Loose miners, not so directed, fall back on whichever of the ubiquitous priority 1s happens to be handiest as they dig out/stairwell/ramp/channel the prior tile (which they may have been working on due to a burrowing that I've just released them from, but at least until they go off to sleep they still somewhat act to continue my priority-1 plans, which is nice).
This is just the way I do it (and haven't yet discussed the Marker Only designations I use, which I really mostly use to pre-plan future expansions, lower-priority even than 7s in the immediate future but may be set up to be 1/2/3-patterned already, ready to be toggle-painted to Standard), and I'm sure there are equally valid alternate methods to plan and micromanage, but this way works best for me.