The Herschel -ATLAS unbiased survey of cold dust in the local Universe is dominated by a surprising population of very blue ( FUV - K < 3.5 ) , dust-rich galaxies with high gas fractions ( { f _ { HI } = M _ { HI } / ( \mbox { $M _ { \ast } $ } + M _ { HI } ) } > 0.5 ) . Dubbed ‘ Blue and Dusty Gas Rich Sources ’ ( BADGRS ) they have cold diffuse dust temperatures , and the highest dust-to-stellar mass ratios of any galaxies in the local Universe . Here , we explore the molecular ISM in a representative sample of BADGRS , using very deep { CO ( J _ { up } = 1 , 2 , 3 ) } observations across the central and outer disk regions . We find very low CO brightnesses ( T _ { p } = 5 - 30 mK ) , despite the bright far-infrared emission and metallicities in the range 0.5 < Z / Z _ { \odot } < 1.0 . The CO line ratios indicate a range of conditions with R _ { 21 } = { T _ { b } ^ { 21 } / T _ { b } ^ { 10 } = 0.6 - 2.1 } and R _ { 31 } = { T _ { b } ^ { 32 } / T _ { b } ^ { 10 } = 0.2 - 1.2 } . Using a metallicity dependent conversion from CO luminosity to molecular gas mass we find \mbox { $ M _ { H 2 } $ } / \mbox { $M _ { d } $ } \sim 7 - 27 and \Sigma _ { H 2 } = 0.5 - 6 \mbox { $ M _ { \odot } $ } { pc ^ { -2 } } , around an order of magnitude lower than expected . The BADGRS have lower molecular gas depletion timescales ( \tau _ { d } \sim 0.5 Gyr ) than other local spirals , lying offset from the Kennicutt-Schmidt relation by a similar factor to Blue Compact Dwarf galaxies . The cold diffuse dust temperature in BADGRS ( 13–16 K ) requires an interstellar radiation field 10–20 times lower than that inferred from their observed surface brightness . We speculate that the dust in these sources has either a very clumpy geometry or a very different opacity in order to explain the cold temperatures and lack of CO emission . BADGRS also have low UV attenuation for their UV colour suggestive of an SMC-type dust attenuation curve , different star formation histories or different dust/star geometry . They lie in a similar part of the IRX- \beta space as z \sim 5 galaxies and may be useful as local analogues for high gas fraction galaxies in the early Universe .