How about this for a counter-example: We take the category

$$ A \stackrel{f_2}{\to} B \rightrightarrows C \stackrel{f_1}{\to} D $$

where there is a unique morphism from $A$ to $C$ and from $B$ to $D$. The only $T_1$ cover is $f_1$ and the only $T_2$ cover is $f_2$.

This category and these topologies satisfy my conditions.

Consider the $T_1$ sheaf which sends every object to a two element set $\underline{1} = ${$0, 1$} and sends $A$ to a singleton set $\underline{0} = ${$0$}, and sends all morphisms to either the identity, or the unique morphism from {$0, 1$} to {$0$}.

$$ \underline{0} \leftarrow \underline{1} \stackrel{id}{\leftleftarrows} \underline{1} \stackrel{id}{\leftarrow} \underline{1} $$

Then the $T_2$ separated presheaf (which happens to already by a $T_2$ sheaf) associated to this $T_1$ sheaf is

$$ \underline{0} \leftarrow \underline{0} \leftleftarrows \underline{1} \stackrel{id}{\leftarrow} \underline{1} $$

which is no longer a $T_1$ sheaf.

How about this for a counter-example: We take the category

$$ A \stackrel{f_2}{\to} B \rightrightarrows C \stackrel{f_1}{\to} D $$

where there is a unique morphism from $A$ to $C$ and from $B$ to $D$. The only $T_1$ cover is $f_1$ and the only $T_2$ cover is $f_2$.

This category and these topologies satisfy my conditions.

Consider the $T_1$ sheaf $F$:

$F(A) = $ a singleton set $\underline{0} = ${$0$}.

$F(B) = $ a two element set $\underline{1} = ${$0, 1$}.

$F(C) = $ a singleton set $\underline{0} = ${$0$}.

$F(D) = \varnothing$ the empty set.

$$ \underline{0} \leftarrow \underline{1} \leftleftarrows \underline{0} \leftarrow \varnothing $$

The two morphisms $F(B) \leftleftarrows F(C)$ are the two distinct morphims $\underline{1} \leftleftarrows \underline{0}$. This is a $T_1$ sheaf.

Then the $T_2$ separated presheaf (which happens to already by a $T_2$ sheaf) associated to this $T_1$ sheaf is

$$ \underline{0} \leftarrow \underline{0} \leftleftarrows \underline{0} \leftarrow \varnothing $$

which is no longer a $T_1$ sheaf.