To get the dotted eighth delay in time in the first place, the easiest way (of course) is to use a delay that has a mode that will give you dotted eighths when you tap quarter notes. If you don't have that facility, the easiest way is probably to count three sixteenths in your head and tap to that beat.
As for staying in time, the first requirement is for your own playing to stay in time with the delay, so that the effect turns out nice and precise. It can help to focus mentally on a slower pulse, from quarter notes to full bars, rather than get distracted by each individual note - don't worry about missing a note here or there, but keep the notes you do play in time.
Secondly, if you're not using a delay that's 'clocked' from an external source, it's important for other players to realise that the delay has to act as the 'master clock' for the band - so the drummer in your band has to follow your delayed part. This may take a bit of practice - it might be worth taking 10 minutes here or there to practice one on one with them.
The more egalitarian solution is to use a master clock source that can drive a click track for the drummer to listen to, and that your delay can also sync to (e.g. via MIDI), but this might be overkill for many situations (and not all delays can do this).
In either of those cases, you have to realise that you're not the boss of the timing any more - let the delay (or click) be the boss, and you stay in time with it, rather than make it stay in time with you. A good practice routine is set the delay time roughly and then get into the habit of syncing your playing to the delay.
If you want to be able to vary the tempo and still keep the synced dotted 8th effect, it helps to be able to re-tap with a minimum of audible artefacts - some delay pedals are better able to do this than others.
If you have enough spare pocket money, a product currently available on the market is the Free The Tone flight time delay - this has a microphone and BPM analyser that will listen to the performance and adjust the delay time automatically...
Though I'm not sure if that would be able to deal with the example you linked to!