We performed $^{75}$As NMR studies on two overdoped high-quality Ba$_{1-x}$K$_{x}$Fe$_2$As$_2$ (x=0.7 and 1.0) single crystals. In the normal states, we found a dramatic increase of the spin-lattice relaxation ($1/^{75}T_1$) from the x=0.7 to the x=1.0 samples. In KFe$_2$As$_2$, the ratio of $1/^{75}T_1TK_n^2$, where $^{75}K_n$ is the Knight shift, increases as temperature drops. These results indicate the existence of a new type of spin fluctuations in KFe$_2$As$_2$ which is accustomed to being treated as a simple Fermi liquid. In the superconducting state, we observe a step-like feature in the temperature dependence of the spin-lattice relaxation of the x=0.7 sample, which supports a two-gap superconductivity as the underdoped materials. However, the temperature scalings of $1/^{75}T_1$ below Tc in the overdoped samples are significantly different from those in the under or optimal doped ones. A power-law scaling behavior $1/^{75}T_1Tsim T^{0.5}$ is observed, which indicates universal strong low energy excitations in the overdoped hole-type superconductors.
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