Two Euler type sums involving binomial coefficient and harmonic number

Question

I want to find the closed forms of the following Euler type sums $$\sum\limits_{n=1}^\infty \frac{H_n}{n^p\binom{2n}{n}}4^n$$ and $$\sum\limits_{n=1}^\infty \frac{H_{2n}}{n^p\binom{2n}{n}}4^n.$$ A similar result please see A Challenging Euler Sum $\sum\limits_{n=1}^\infty \frac{H_n}{\tbinom{2n}{n}}$. By Mathematica,
the following equations seems to be true $$\sum\limits_{n=1}^\infty \frac{H_n}{n^2\binom{2n}{n}}4^n=\pi^2\log(2)+\frac{7}{2}\zeta(3)$$ and $$\sum\limits_{n=1}^\infty \frac{H_{2n}}{n^2\binom{2n}{n}}4^n=\frac{1}{2}\pi^2\log(2)+\frac{35}{4}\zeta(3).$$ How to find their closed forms for general $p=0,1,2,3,\ldots$?