Min weighted edge cover: don't follow proof in Schrijver
Let $M$ be a maximal subset of $F'$ which is a matching. Any vertex $v$ not covered by $M$ must be covered by some edge $e_v=(v,w)$ in $F'$. Since $M$ is maximal, $w$ must be covered by $M$. It follows that if $v_1 \neq v_2$ are not covered by $M$, then $e_{v_1} \neq e_{v_2}$.
The matching $M'$ consists of the two copies of each edge in $M$, and of the edges $(v,\tilde{v})$ for each $v$ not covered by $M$. Edges of the first type have total weight $2w(M)$. Since $w(v,\tilde{v}) \leq 2w(e_v)$, edges of the second type have total weight at most $2w(F' \setminus M)$ (this crucially uses that $e_{v_1} \neq e_{v_2}$ for any $v_1 \neq v_2$ not covered by $M$). Hence $w(M') \leq 2w(F')$.