Thinking of D Dorian as having the "same notes" as C major is true in absolute terms. But most people hear music relatively to the tonal centre of gravity of the piece. The notes D, E, F, G, A, B, C have completely different relationships to a D tonic than they do to a C tonic, so D Dorian shouldn't sound like C major at all.
One practical step to take might be to just have a good listen to a few songs and solos in the Dorian mode, and get an idea for how it sounds and feels completely different to major. What is a good solo to learn in the Dorian mode? has a few suggestions for solos, and there are lots of web pages with suggestions of Dorian songs. Oye Como Va (Santana), Wicked Game (Chris Isaac), and Scarborough Fair (trad./Simon & Garfunkel) are often mentioned. I'm not sure they all stick rigidly to Dorian - doing that analysis yourself might be a good exercise.
With its minor/flattened third, playing in D Dorian should feel quite like playing in D minor. in fact you could think of the Dorian mode as being like the 'melodic' minor scale, but always with the 'ascending' 6th, and the 'descending' 7th. If you use all the wiggle room in the concept of minor tonality, maybe you don't 'need' Dorian at all!
… Or it may just be that thinking of Dorian is fine, but the idea of it coming from a major 'parent scale' isn't helping you. Perhaps it would be better just to think of Dorian as a scale pattern in its own right? Personally, I don't think of the modes as coming from a major parent scale - I think of the diatonic scale (i.e. the note pattern, without any particular root) as the "parent" concept, and then of major/minor tonality and modal tonality as both being concepts derived from that.