In a word yes!
The whole point of the minor blues scale notes is that they produce a 'sweet and sour' effect against the major scale notes. So out of tune that they sound (sort of) right!
What happens with good blues players is that sometimes they bend the m3 up to a M3; although they also bend it half-way, and the listener 'hears' it go to M3, which, of ...
Playing blues on a harp, drawing is the way to go. Blowing on, say, a C harp will give you the notes of that chord - C. So the blues notes, mainly, give the chord a fourth away - G. So, for a song in G, you'd need a C harp.
To calculate what you need, know the guitar key, and count backwards 5, or more simply, forwards 4. As in, song's in A, use D. Song's ...
Generally, for blues, if guitar plays in C you would use an F harp. This is to get the Bb ("blue 7th"), otherwise known as Mixolydian mode. In general, you want the harp whose key is a perfect 4th above the key of the blues. So:
For a E blues, an A harp
For a D blues, a G harp
and so on.
It sounds kind of OK, but it is not in the blues idiom to my ears. One reason is that you are filling in all the elements of the chords. In jazz/blues playing it is really common to omit the fifth from a lot of these chords, and start from what is called a "shell" voicing, which contains the root, third and seventh (whether dominant or maj7) and then ...