What string theory aims to do is to unify GR with quantum field theory, not quantum mechanics. Before discussing what string theory does, we need to correct some misconceptions you have.
I’m not sure what you mean by time “varying” vs “being a constant”; I assume you mean that in quantum mechanics time is a parameter and space is an operator, and we don’t have Lorentz invariance. However, that comes up when we compare QM with special relativity too, and led to the development of quantum field theory, which fully incorporates special relativity.
Quantum mechanics solves for the behaviour of a system composed of a certain number of particles by finding the eigenstates of a Hamiltonian operator. Quantum field theory does something completely different. Why? Because quantum mechanics quantises classical mechanics, whereas classical field theory quantises classical field theory. Forget everything you know about quantum theory for the moment and just compare these two equations: $$\ddot{\mathbf{x}}=-\boldsymbol{\nabla}V,\, \partial_\mu F^{\mu\nu}=\mu_0 j^\nu.$$Both are equations of motion. The first can be obtained as an Euler-Lagrange equation of the action $$S=\int dt\left( \frac{1}{2}m\dot{x}^2-V\left( x\right) \right),$$while the second can be obtained as one of several Euler-Lagrange equations of the action $$S=\int d^4x \left(-\frac{1}{4\mu_0}F_{\mu\nu}F^{\mu\nu}-A_\nu j^\nu\right)$$where we abbreviate $d^4x=dtd^3x,\,d^3x=dtdxdydz$. Just as classical mechanics writes an action in terms of functions of time called coordinates and time derivatives thereof (and possibly also time itself, in which case energy is not conserved), so classical field theory writes an action in terms of functions of spacetime called fields and spacetime derivatives thereof (and possibly also spacetime itself, in which case energy is not conserved if in particular there is an explicit time dependence).
What does all this have to do with the Hamiltonian being ditched? Well, when we turn quantisation back on again we discover we can no longer have a probability amplitude for one particle's location. If you rearrange the time-dependent Schrödinger equation as $\dot{\psi}=i\left( \frac{\hbar}{2m}\nabla^2-\frac{V}{\hbar} \right)\psi$, you can use $\rho=\psi^\ast \psi$ to prove that $\dot{\rho}+\boldsymbol{\nabla}\cdot\mathbf{j}=0$ for probability 3-current $\mathbf{j}=\frac{i\hbar}{2m}\left(\psi\boldsymbol{\nabla}\psi^{\ast}-\psi^{\ast}\boldsymbol{\nabla}\psi\right)$. If this probability interpretation can survive in relativity, we need $\partial_\mu j^\mu = 0$ with $\int d^3 x j^0 = 1$ for some $j^\mu$ expressible in terms of solutions of a relativistic variant of the Schrödinger equation. But in theory, the Schrödinger equation can be interpreted as an equation in some field $\psi$ that has no probabilistic interpretation. Indeed, any relativistic upgrade ends up with solutions for which $\int d^3 x j^0 \le 0$. The ultimate resolution is to see $j^0$ as a difference bewteen particle and antiparticle densities and take the fields in the theory as descriptions of the entire population of such particles and antiparticles in the universe (e.g. the Dirac spinor describes all electrons and positrons). But if particles are now measurable properties of fields in the same way we're used to thinking of momentum as a measurable property of one particle, it is the Lagrangian, not the Hamiltonian, that takes centre stage. The theory doesn't need energy conservation to work, but it gets it anyway in Minkowski space (or, indeed, any spacetime whose metric tensor's determinant is time-independent).
Now let's talk about renormalisation. All current field theories are low-energy variants, obtainable as discussed here, of as yet unknown high-energy theories. As explained here, the proof that general relativity isn't a renormalisable quantum field theory boils down to showing that, whereas any renormalisable QFT has entropy-energy relation $S\propto E^{1-d^{-1}}$ at high energies in a $d$-dimensional spacetime, for GR we get a different power law. Which one? It depends on the spacetime geometry considered. The high-energy spectrum is that of a large black hole, which when plonked into an otherwise large-scale Anti de Sitter spacetime gives $S\propto E^{1-\left( d-1\right)^{-1}}$, so the high-energy conformal field theory recovered is missing a dimension. The AdS/CFT correspondence often discussed in string theory places this CFT on the boundary of spacetime. (The observed acceleration of the universe's expansion indicates de Sitter space is a better model of our universe, but there is also a dS/CFT correspondence that string theory can use.) You may have heard people talk of a "holographic principle"; this is what they're talking about.
The paper I linked to first in the above paragraph ends with a brief clarification of some common misconceptions about exactly what is "wrong" with GR. It'd be better to say there are some unusual features of it, so that we have slightly different unanswered questions hanging over it. Although GR isn't renormalisable, we can make some low-energy predictions for its quantisation, e.g. a calculable $r^{-3}$ correction to Newton's potential. We can also predict that, at or before the Planck energy, new high-energy physics must take over.
String theory attempts to be a high-energy theory whose low-energy spectrum recovers all known physics. Instead of modelling particles as point masses, string theory considers them to have some length, which addresses the renormalisability of gravity. String theory requires a number of extra spacetime dimensions to work, but these are invisible at low energies because they're "compactified", i.e. small in extent. If the size of these new dimensions was the Planck length, new physics would occur at the Planck energy. The energy cutoff for new physics is actually a little lower than the Planck energy, because string theory responds to the hierarchy problem (i.e. gravity being much weaker than the weak interaction) by positing compactified dimensions larger than the Planck length.
String theorists still don't know why several dimensions would compactify, or why having done so they would adopt the specific geometry (called a Calabi-Yau manifold) that they did. But string theory predicts that the topology of this manifold determines the laws of physics (including the set of particle species), while the sizes of the holes in this topology determine the parameters in those laws. This can create about 10^500 possible kinds of physics, forming what's often called the string theory landscape. Pinpointing our physics therein is still an ongoing research topic.
If you'd like to learn more about string theory's approach, see here.
