We consider the \emph{$k$-edge connected spanning subgraph} (kECSS) problem, where we are given an undirected graph $G = (V, E)$ with nonnegative edge costs $\{c_e\}_{e\in E}$, and we seek a minimum-cost \emph{$k$-edge connected} subgraph $H$ of $G$. For even $k$, we present a polytime algorithm that computes a $(k-2)$-edge connected subgraph of cost at most the optimal value $LP^*$ of the natural LP-relaxation for kECSS; for odd $k$, we obtain a $(k-3)$-edge connected subgraph of cost at most $LP^*$. Since kECSS is APX-hard for all $k\geq 2$, our results are nearly optimal. They also significantly improve upon the recent work of Hershkowitz et al., both in terms of solution quality and the simplicity of algorithm and its analysis. Our techniques also yield an alternate guarantee, where we obtain a $(k-1)$-edge connected subgraph of cost at most $1.5\cdot LP^*$; with unit edge costs, the cost guarantee improves to $(1+\frac{4}{3k})\cdot LP^*$, which improves upon the state-of-the-art approximation for unit edge costs, but with a unit loss in edge connectivity. Our kECSS-result also yields results for the \emph{$k$-edge connected spanning multigraph} (kECSM) problem, where multiple copies of an edge can be selected: we obtain a $(1+2/k)$-approximation algorithm for even $k$, and a $(1+3/k)$-approximation algorithm for odd $k$. Our techniques extend to the degree-bounded versions of kECSS and kECSM, wherein we also impose degree lower- and upper- bounds on the nodes. We obtain the same cost and connectivity guarantees for these degree-bounded versions with an additive violation of (roughly) $2$ for the degree bounds. These are the first results for degree-bounded \{kECSS,kECSM\} of the form where the cost of the solution obtained is at most the optimum, and the connectivity constraints are violated by an additive constant.