Let X X be an abstract not necessarily compact orientable CR manifold of dimension 2 n − 1 2n-1 , n ⩾ 2 n\geqslant 2 , and let L k L^k be the k k -th tensor power of a CR complex line bundle L L over X X . Given q ∈ { 0 , 1 , … , n − 1 } q\in \left \{0,1,\ldots ,n-1\right \} , let ◻ b , k ( q ) \Box ^{(q)}_{b,k} be the Gaffney extension of Kohn Laplacian for ( 0 , q ) (0,q) forms with values in L k L^k . For λ ≥ 0 \lambda \geq 0 , let Π k , ≤ λ ( q ) := E ( ( − ∞ , λ ] ) \Pi ^{(q)}_{k,\leq \lambda }:=E((-\infty ,\lambda ]) , where E E denotes the spectral measure of ◻ b , k ( q ) \Box ^{(q)}_{b,k} . In this work, we prove that Π k , ≤ k − N 0 ( q ) F k ∗ \Pi ^{(q)}_{k,\leq k^{-N_0}}F^*_k , F k Π k , ≤ k − N 0 ( q ) F k ∗ F_k\Pi ^{(q)}_{k,\leq k^{-N_0}}F^*_k , N 0 ≥ 1 N_0\geq 1 , admit asymptotic expansions with respect to k k on the non-degenerate part of the characteristic manifold of ◻ b , k ( q ) \Box ^{(q)}_{b,k} , where F k F_k is some kind of microlocal cut-off function. Moreover, we show that F k Π k , ≤ 0 ( q ) F k ∗ F_k\Pi ^{(q)}_{k,\leq 0}F^*_k admits a full asymptotic expansion with respect to k k if ◻ b , k ( q ) \Box ^{(q)}_{b,k} has small spectral gap property with respect to F k F_k and Π k , ≤ 0 ( q ) \Pi ^{(q)}_{k,\leq 0} is k k -negligible away the diagonal with respect to F k F_k . By using these asymptotics, we establish almost Kodaira embedding theorems on CR manifolds and Kodaira embedding theorems on CR manifolds with transversal CR S 1 S^1 action.