Let X be an abstract not necessarily compact orientable CR manifold of dimension 2n − 1, n > 2, and let L be the k-th tensor power of a CR complex line bundle L over X. Given q ∈ {0, 1, . . . , n− 1}, let (q) b,k be the Gaffney extension of Kohn Laplacian for (0, q) forms with values in L. For λ ≥ 0, let Π (q) k,≤λ := E((−∞, λ]), where E denotes the spectral measure of (q) b,k . In this work, we prove that Π (q) k,≤k−N0 F ∗ k , FkΠ (q) k,≤k−N0 F ∗ k , N0 ≥ 1, admit asymptotic expansions on the non-degenerate part of the characteristic manifold of (q) b,k , where Fk is some kind of microlocal cut-off function. Moreover, we show that FkΠ (q) k,≤0F ∗ k admits a full asymptotic expansion if (q) b,k has small spectral gap property with respect to Fk and Π (q) k,≤0 is k-negligible away the diagonal with respect to Fk. By using these asymptotics, we establish almost Kodaira embedding theorems on CR manifolds and Kodaira embedding theorems on CR manifolds with transversal CR S1 actions.