Getting that "The War on Drugs" guitar tone

Hey,

So I've been working on recording a song and I am trying to get a solo tone similar to the kind of long, sustained, gain-y yet clean sound that The War on Drugs tends to use.

Any suggestions? I'm mostly recording direct input through my pedal board into a tascam sound card. I've fiddled around with my overdrive pedal (Soul Food) but the tone gets too distorted and muddy to give that clean sustain.

Is it time to get a sustain pedal? Apologies in advance if what I'm asking about gear doesn't make much sense..