I'll use $x,y$ for your $t_1,t_2$, and $h_1,h_2,h_3$ for your three formulas for $h$.
Now if $(6k-1,6k+1)$ is a pair of twin primes, then the product $36k^2-1$ should have only two prime factors. If the algebraic factorizations of these turn out to factor into three (or more) terms all clearly greater than 1, then you have what you want.
We have $36h_1^2-1=(5+6x)(7+6y)(36xy+42x+30y+37)$. So $h_1$ does what you want.
Also $36h_2^2-1=(5+6x)(5+6y)(36xy+30x+30y+23)$. So $h_2$ does what you want.
I'll leave you to do $h_3$...
EDIT: We can find other examples of such $h$ by the following method. Choose $a,b$ such that $ab=6k+1$ [$6k-1$ would also work]. Now compute $(6x+a)(6y+b)=6(6xy+bx+ay+k)+1.$ So if we define $h=6xy+bx+ay+k$ we will have $6h+1=(6x+a)(6y+b)$ and $6h-1=36xy+6bx+6ay+6k-1$. Thus with positive $x,y$ we can't have $6h+1$ and $6h-1$ both prime, since here we have in fact $6h+1=(6x+a)(6y+b)$ which is not prime.
Added note: In the above I assumed $x,y>0$. If one allows negative $x,y$ there may be a way to make one of the factors 1, while the other two are integer primes, by the latter meaning we allow $-2,-3,-5,...$ also as primes.