Solution:

The diagonal is $DB = \sqrt{4^2 + 3^2} = 5.$

Right triangles $EBA,$ $DBC,$ and $BFC$ are all similar to $\triangle DBA.$ From $\triangle EBA,$ $$\dfrac{EB}{AB} = \dfrac{DB}{BC} \implies \dfrac{EB}{4} = \dfrac{5}{3} \implies EB = \dfrac{20}{3}.$$

From $\triangle BFC,$ $$\dfrac{BF}{BC} = \dfrac{DB}{AB} \implies \dfrac{BF}{3} = \dfrac{5}{4} \implies BF = \dfrac{15}{4}.$$

Therefore $$EF = EB + BF = \dfrac{20}{3} + \dfrac{15}{4} = \dfrac{125}{12}.$$

Thus, the correct answer is C.