even states

psiII[x_]=Cosh[κ x];
psiIII[x_]=AA Sin[k(x-((Δ/2)+a))];

Solve[psiII[(Δ/2)]==psiIII[(Δ/2)],AA]//Flatten

psiIIIe[x_]=-Cosh[((Δ κ)/2)] Csc[a k] Sin[k(x-((Δ/2)+a))];

psiII'[(Δ/2)]==psiIIIe'[(Δ/2)]

κ Tanh[((Δ κ)/2)]==-k  Cot[a k]

setting k=π/a-δk, Taylor series of rhs

Series[-((π/a)-δk) Cot[a ((π/a)-δk)],{δk,0,1}]

Solve[κ Tanh[((Δ κ)/2)]==(π/((a^2) δk))-(1/a),δk]

E0((( (((π/a)-(π/((a^2) κ Tanh[((Δ κ)/2)])))/((π/a)))))^2)//Expand

Ee=E0-((2 E0 Coth[((Δ κ)/2)])/(a κ))

psie[x_]:=(Piecewise[{{psiIIIe[Abs[x]], Abs[x]>(1/2)}, {Cosh[κ x], True}}])

for odd states, simply replace Tanh->Coth.  Strong tunneling, aκ>>1

psiIIIo[x_]=-Sinh[((Δ κ)/2)] Csc[a k] Sin[k(x-((Δ/2)+a))];

psio[x_]:=(Piecewise[{{Sign[x]psiIIIo[Abs[x]], Abs[x]>(1/2)}, {Sinh[κ x], True}}])

Eo=E0-((2 E0 Tanh[((Δ κ)/2)])/(a κ))

Eo-Ee//TrigToExp//Simplify

which is approximately

((-8 (E^(Δ κ)) E0)/(-a (E^(2 Δ κ)) κ))

Plot[psie[x]//.{Δ->1,a->1,k-> (π/a)-((π Coth[((Δ κ)/2)])/((a^2) κ)),κ->30},{x,-1.5,1.5}]

Plot[psio[x]//.{Δ->1,a->1,k-> (π/a)-((π Tanh[((Δ κ)/2)])/((a^2) κ)),κ->30},{x,-1.5,1.5}]

psir=(psie[x]//.{Δ->1,a->1,k-> (π/a)-((π Coth[((Δ κ)/2)])/((a^2) κ)),κ->30})+psio[x]//.{Δ->1,a->1,k-> (π/a)-((π Tanh[((Δ κ)/2)])/((a^2) κ)),κ->30}

psil=(psie[x]//.{Δ->1,a->1,k-> (π/a)-((π Coth[((Δ κ)/2)])/((a^2) κ)),κ->30})-psio[x]//.{Δ->1,a->1,k-> (π/a)-((π Tanh[((Δ κ)/2)])/((a^2) κ)),κ->30}

Animate[Plot[(((psir Cos[th]+psil Sin[th]))^2),{x,-1.5,1.5},PlotRange->{0,1.2 (10^15)}],{th,0,2π}]