In order to get a shiny Pokemon, you have to find a strong foe dungeon, and at some point in the dungeon, one of the Pokemon listed below will appear, and it has a slim chance of being shiny when doing so. They can appear multiple times in the dungeon, and there's no way to increase the chances of them being shiny. If they are shiny, I'd suggest you save, defeat it with the LEADER of your team and make sure to have a Friend Bow equipped (it increases the chance of them joining you) and hope it joins your team, if it doesn't, power off and on the game, it should pop up with the Pokemon being shiny, since you saved while it was shiny. As for soft resetting, the Pokemon has a chance to appear at every time you go down a stair, so I'd suggest saving right before a stair, and then going down, this could take a long time, because they don't appear every time you go down the stair, and it won't be shiny every time either.
Pokemon that you can find shiny and where they are:
Butterfree, found in the Howling Forest
Ninetales, found in the Northern Range
Rapidash, found at Mt. Steel
Electrode, found at Mt. Faraway
Exeggcute, found at Wyvern Hill
Exeggutor, found in the Pitfall Valley
Starmie, found in the Grand Sea
Magikarp, found in the Marvelous Sea
Gyarados, found in the Waterfall Pond
Ditto, found at the Buried Relic
Dragonite, found at Mt. Thunder
Hoothoot, found in the Murky Cave
Noctowl, found in the Lapis Cave
Ampharos, found in the Sinister Woods
Azumarill, found at Mt. Blaze
Sudowoodo, found in the Desert Region
Aipom, found in Solar Cave
Espeon, found at Remains Island
Wobbuffet, found at the Darknight Relic
Shuckle, found in the Southern Cavern
Corsola, found in the Far-Off Sea
Celebi, found in the Mystery Houses once you recruit a Celebi in Purity Forest, a Shiny Celebi may appear in M Houses
Spinda, found in Frosty Forest
Altaria, found at the Fantasy Strait
Zangoose, found in the Western Cave
Metagross, found in the Silent Chasm
Ambipomm found in the Great Canyon
Lickilicky, found at Mt. Freeze
And that's how you get a shiny in Rescue Team DX, and YES soft reseting works, but doesn't increase the chance, steps for finding one of these shinies are at the top
Source: Knowledge and https://www.serebii.net/dungeonrescueteamdx/shinypokemon.shtml
