Don't see 10^29 or 10^27 in the attached notebook plots so it is not at all clear what you are trying to accomplish. What is a "perfect linear scale"?

A single scale factor cannot be used since the range of the plots is different. One way to do it e.g. for the first plot

Plot[{Im[y] /. {xi -> 10}, Re[x] /. xi -> 50, Re[x] /. xi -> 100}, {k,0, 100},
 PlotRange -> All,
 PlotStyle -> {Directive[Red, Thick], Directive[Blue, Dashed, Thick], Directive[Black, Thick, Dashed]},
 Frame -> True,
 FrameStyle -> Thick,
 PlotLegends -> {"K for \[Xi] = 10", "K for \[Xi] = 50", "K for \[Xi] |= 100"}, 
 FrameLabel -> {"Wave number (K)", "Imaginary frequency (\!\(\*SubscriptBox[\(\[CapitalOmega]\), \ \(i\)]\))"},
 RotateLabel -> True,
 AspectRatio -> 0.5,
 FrameTicks -> {{Charting`ScaledTicks[{10^9 # &, #/10^9 &}], None}, {Automatic, None}}]