We measure by experiment and particle-based simulation the rheology of concentrated, non-Brownian droplet emulsions functionalized with surface-bound single-stranded (ss), "sticky," DNA. In the absence of ssDNA, the emulsion viscosity increases with the dispersed phase volume fraction ϕ, before passing through a liquid-solid transition at a critical ϕ_{c} related to random close packing. Introducing ssDNA leads to a liquid-solid transition at ϕ<ϕ_{c}, the onset being set by the droplet valency N and the ssDNA concentration (or simulated binding strength ε). Using insight from simulation, we identify three key behaviors: (i) jammed suspensions (ϕ>ϕ_{c}≈0.64) show weak effects of functionalization, with elastic rheology instead governed by droplet stiffness; (ii) suspensions with ϕ<ϕ_{c} and N=1, 2 always exhibit viscous rheology, regardless of functionalization; and (iii) for ϕ<ϕ_{c} and N>3, functionalization leads to a controllable viscous-elastic transition. We present state diagrams showing the range of rheological tuning attainable by these means.