Resolving a double-point singularity in terms of Morse theory

Question

Background: I want to understand the "homology realization problem", a.k.a. Steenrod's problem, in 4-dimensional case. The precise statement that I'm considering is:

Theorem. Any 2-dimensional homology class $\alpha \in H_2(M;\mathbb{Z})$ of a (smooth) closed 4-manifold $M$ is represented by a (smooth) submanifold $S\subset M$, i.e., $\iota_*[S]=\alpha$ where $\iota : S\to M$ is the inclusion.

(All manifolds are assumed to be oriented for brevity.)

There are largely two proofs of this theorem that I know of; One uses the Brown representability $H^2(M;\mathbb{Z})=[M,K(\mathbb{Z},2)]$ with the Poincare duality. I have a concrete understanding on it. The other proof which I'm going to elaborate on below is where I was stuck on.

You can first choose a continuous function from $S$ into $M$ which realizes $\alpha$: say $f:S\to M$ is one such function. Then, by a simple transversality argument, we can homotope $f$ to be an immersion, which necessarily has only transverse double-point singularities. These double-points can be removed at the price of increasing the genus of $S$ as follows:

Locally the self-intersection of $S$ near a double-point is given as the equation $z_1z_2=0$ (in $\mathbb{C}^2\cong \mathbb{R}^4$). Then you can replace this chart with perturbing the equation a little bit: $z_1z_2=\epsilon$ for some nonzero $\epsilon$. By replacing all the singularity in this fashion, we get an embedded surface which realizes $\alpha$.

My question(s) is:

1. I feel a strong analogy between this argument and the standard Morse-theoretic argument which constructs a bordism between $g^{-1}(-\epsilon)$ and $g^{-1}(\epsilon)$ when $0$ is a critical value of the Morse function $g\in C(M)$. Especially, locally the replacement is exactly when $g(z_1,z_2)=z_1z_2$, and if the standard Morse-theoretic argument can be applied(or modified) to obtain a bordism between the immersion $S\to M$ and the embedded surface at the end, then clearly the embedded surface realizes $\alpha$ which concludes the theorem. Is this true?
2. If this is not the case, I don't understand why this replacement process does not vary the homology class of $S$. Of course, if we have the Morse-theoretic argument which generates a bordism as in Q1, then clearly the homology class does not vary, but I'm interested in what extent this replacement is valid: Is it true that a singularity of type $f(x)=0$ can always be desingularized by replacing the part with $f(x)=\epsilon$?