Yep, you have to babysit health bars. Keep an eye on your PW:S targets' health bars to notice when the shield has been consumed/has expired, leaving the player vulnerable damage. Watch the Weakened Soul icon on your PW:S targets' health bars to know when you can re-apply PW:S, as warranted. You have to keep track of where your PoM is sitting (and any other priest's PoM because if you cast yours on someone who has another priest's PoM, the other priest's PoM vanishes), to know how many charges it has left, so you're re-applying it appropriately.
But yep, you also have to keep an eye on the DBM timers & warnings, especially as a Disc priest, since your focus is on not allowing the health bars to drain in the first place, by judicious application of PW:S and SS.
Once you hit end-game healing, I recommend opening the DBM config to adjust the warnings & timers to show only what you want them to show. I mean, you could do that now too, except that you're still ripping through content with new 5mans appearing on a lvl-by-lvl basis, and I'd presume it'd seem like an awful lot of work. It's definitely worth it for raiding Disc priests, though.
You mention that you've done a bunch of UI work, so perhaps you've already covered the following points. I'll list them anyway, in case there's something you hadn't thought of which seems like it could help.
1) Raid Frames
Get used to setting your raid frames on the small size (while still being able to make out the buff/debuff icons which appear on them, of course). It's not of as much importance while you're only doing 5mans, but you really *do* want to be able to see as much of the screen as you possibly can for raid awareness. Also, raid frames can get in the way of trying to place PW:B (or any other spell placed with a reticule).
Consider placing your raid frames directly under your toon's feet, if you don't already have them there. While they're more in the way of reticule-placement there, that's far less of a problem then having them "nicely tucked away" at the far edge of your screen, where you sometimes have to scramble to get your mouse over to them in the middle of ongoing mid-fight chaos. Additionally, placing raid frames under your feet allows you to more easily monitor health bars, your ability cooldowns, *and* whether you're standing in fire... all at the same time.
If you haven't already, consider going into your raid frame config to select the "only show dispellable debuffs" option. It doesn't make a huge difference on many fights, but it can really help on some fights.
Once you have your raid frame position decided upon, go into the DBM config again and play around with where you want those impending timers to show up. The default positioning might not be what you want.
3) Atonement macros
#showtooltip Holy Fire
/use [@targettarget, harm]  Holy Fire
... make another one for Smite (substituting "Smite" wherever you have "Holy Fire" in the first macro, of course).
I only have a very basic understanding of macro creation, so I can't advise how to set that up as a mouseover macro, though I'm certain it's possible. If no one else responds about that here, set up another thread to ask that question specifically... or head over to the battlenet macro & UI forum.
... if you haven't worked with macros much yet, the explanation of what this macro does is:
- if you have a hostile target, it will cast Holy Fire.
- if you have a friendly target, it will cast Holy Fire on that player's hostile target.
... you run into a few situations now and then where your friendly target doesn't have a hostile target targeted (ie... you have a healer targeted, so their target isn't hostile... or you have someone targeted who doesn't currently have anything targeted) so it's good to have the macro have the functionality of using Holy Fire "normally" on a hostile target.
Always find excuses to blow your cooldowns, even if the run is going smoothly. If you don't always make a habit of looking for situations to use them (even if only for fun!) you won't remember to use them when you need them (you'll just try to heal harder, and it won't be enough). Practice never letting your stacks of Evangelism drop (hint: Holy Fire on cooldown is excellent Disc practice). Practice popping Archangel now and then. Definitely drop PW:B. Delight in Spirit Shell keeping health bars at 100% through AoE damage (this also keeps you in the practice of choosing good PoH targets... a good PoH target is someone who is within 40 yards of everyone else in their party... don't choose the mage standing way off to the right as the target to center your PoH on). And *definitely* pop Pain Suppression.
But yeah, it's just practice. Lots and lots and lots of practice. Especially if you've been playing one role for quite a while, healing will seem very alien.