In my opinion, one way to practice the modes is to write new music with them. Like TimCox touches on above, each mode has it's "flavor" tone(s). Dorian, for instance, has that major 6th; otherwise, it is just a minor scale. So, it's entirely possible to write something in natural minor (Aeolian mode) and have it be indiscernible from Dorian mode if you never hit that major 6.
So I think it's important to focus on those "flavor" tones, and maybe craft some new lines that really emphasize them. As you know, the major scale modes are just starting a major scale from a point other than the tonic. But as the Dorian vs. Aeolian example illustrates, it becomes entirely about context. If you aren't working those "flavor" tones, then you aren't really using the modes. So you just have to drill into your head which tones give which mode its flavor.
Also, I like to go to something like YouTube and search for (e.g.) "G Mixolydian jam track" or whatever and just play along to it til I'm sick of it. The user "Quist" has a bunch of these tracks with different keys and modes, for instance. A lot of them seemed aimed at guitar players, but work just as well for any type of instrument, provided you don't mind playing along to whatever genre they happen to be in. This way, you can practice learning a certain mode/key, as well as practicing your improvisational chops and your ability to adapt to different genres/styles, and you also get a feel for which harmonic context is appropriate for which mode and stuff like that.
It's also worth noting how the modes and keys relate to one another. For a simplistic example, if you're trying to play D Dorian, you can basically let your fingers fall into those comfortable C Major positions, and simply adjust your mental contextualization of that familiar pattern. Same with E Phrygian, F Lydian, etc, they're all those same C Major positions you already drilled in. So if you think about it like this, once your fingers are comfortable with each major scale, then they are (by extension) comfortable, to an extent, with all the modes of that particular key, provided you can learn to flip that mental switch.
Your mileage may vary, but I find this helps my fingers get a feel for each mode, without getting into that whole boring rote memorization tactic of just playing the modes up and down by themselves. But I'm no instructor or professional by any means, and everyone is different.