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/(M*+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 (Tp=15-30 mK), despite the bright far-infrared emission and metallicities in the range 0.5<Z/Z_sun<1.0. The CO line ratios indicate a range of conditions with R_21=0.6-2.1 and R_31=0.2-1.2. Using a metallicity dependent conversion from CO luminosity to molecular gas mass we find M_H2/M_d=7-27 and Sigma_H2=0.5-6 M_sun pc^-2, around an order of magnitude lower than expected. The BADGRS have lower molecular gas depletion timescales (tau_d = 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 temperatures in BADGRS (13-16 K) require 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=5 galaxies and may be useful as local analogues for high gas fraction galaxies in the early Universe.
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