Strings do make a difference to sound but not enough so that we can recommend a particular brand for jazz or a particular brand for blues. Blues and jazz players tend to use a heavy gauge string to get a big mellow sound but there are several reasons not to emulate this;
1) Heavy gauge strings are real hard on the fingers. Some will tell you this builds finger strength but I reckon that's like your gym teacher saying press-ups are character building.
2) Jumping up several gauges needs the intonation adjusting, maybe the action as well and on some guitars it may strain and bow the neck needing truss rod relief adjustment!
You can get a jazz blues sound simply by working your guitar controls. Try this;
1) Turn the volume to zero and the tone full up (most trebly).
2) Turn your amp full up, strike a guitar string normally and then raise the guitar volume knob until you first hear it ring. You now have the 'bit-point', keeping in this position gives an expressive sensitivity. It varies by amp and guitar but there's always a bite-point.
3) Let some strings ring and roll off the tone knob. You'll hear the sound get less trebley. There's a point where the sound suddenly goes muddy. Find this and then rise a bit away from this. It should be mellow without being mushy. Perfect for jazz/blues.
'The sound of the city seems to disappear'