The size and shape of the compact dimensions are dynamical properties anyway, even before you consider complicated effects like flop transitions. Are you familiar with the idea of a closed universe in general relativity, and how it can expand or contract? It's the exact same phenomenon. The radius of an extra dimension in string theory can increase or decrease dynamically, and this is a major, major issue for the particle physics side of string theory, because the particle properties (masses, charges, etc) are supposed to be set by the extra dimensions, and so the extra dimensions have to be geometrically stable somehow. The relevant properties (size and shape of the extra dimensions, location of branes in the extra dimensions) are called "moduli", and this is the issue of "moduli stabilization": If the moduli changed, particle physics would change drastically, and that contradicts the observed regularity of physical law, not just in experiments on Earth, but as observed on cosmic scales of time and space. So for the time since the big bang, and at least across the billions of light-years of the observed universe, some effect has to be keeping the moduli stable.
The main idea (though not the only one) is that there are electromagnetism-like fields which take nonzero values all the way around some of the extra dimensions. Under the right conditions, they will stabilize the size of the extra dimensions at a finite radius/area/volume, preventing them from expanding to infinity or collapsing to zero. These scenarios are studied under the name of "flux compactifications".
You talked about strings changing size when the extra dimensions change size too. Here we need to distinguish between a string as something that moves through a space, and strings as objects which collectively make up the space. The first part is the easier part of string theory (relatively speaking). The properties of the space are treated as a fixed background independent of the strings, and then you just study how the strings move and interact in that background.
As for the second part, that's probably best described by string field theory, which has a lot of unsolved problems. But let me at least say how I think about this. An individual string has many distinct possible states of excitation. For the closed string, one of these is the graviton. In quantum field theory, there is a standard way to build up a field state as a superposition of many particle states. Basically, the classical waves in the field are built up from the wavefunctions of many many particles. Ever since Einstein, gravity is described geometrically, by the metric tensor, but it is also a type of field, and so the variations in the metric tensor (gravity wells, gravitational waves, etc) can be understood in this way as coherent superpositions of many "graviton wavefunctions". The quantum theory of gravitons as point particles is problematic, but that was the big attraction of string theory - the extra detail of stringlike internal structure inside all "particles" ameliorated the problems.
So, the geometry of space, as described by the metric, ought to be understood as a big coherent quantum sum of closed-string graviton states. It is a little as if there were closed strings everywhere in space, but because it's a quantum sum, it's a statement about probabilities of what the closed strings are doing everywhere. I also want to add that the electromagnetism-like fields appearing in the flux compactifications also derive from the closed strings - in the same way, but in a different sort of excited state, not a graviton state.
In other words, the background through which strings move in the "easy" version of string theory (perturbative string theory) ought itself to have a description in terms of strings, in the "hard" version of string theory (string field theory or some other approach). The metric of the extra dimensions, the "fluxes", the "branes", can all be expressed as quantum sums of very large numbers of strings, exhibiting coherent collective behavior. And you can go a long way towards realizing this ambition, but not all the way, not yet anyway. So what I'm saying is that if you ask, how do the strings change when the extra dimensions change, that's a very different question for a string in the "foreground" and a string in the "background".
I'd also better emphasize that a lot of what I'm saying, comes from a general understanding of how this all works, rather than from intimate acquaintance with the specific technical problems of representing branes in string field theory, and so on. There may be someone out there who can speak more accurately and authoritatively about the deeper issues.
P.S. And speaking of deeper issues, thinking about this question made me realize a very basic deficiency in my own understanding, to do with
https://www.physicsforums.com/showthread.php?t=419450&page=21#post2934361". T-duality swaps radius R for radius 1/R, and swaps the (quantized) momentum of the string along the closed direction with the winding number of the string along that direction - OK. But it hadn't quite dawned on me what this means for space-time distance. Yes, we can interpret this to mean that there's a smallest distance, because when R is less than "one", it's equivalent to a situation in a dual theory with R greater than "one". But how exactly do you translate that intuition into something at the level of operators and measurements? Is there a T-duality-invariant, operational definition of length which applies on both sides of the duality? It seems like it should relate to what I've called the hard part of string theory, where you understand everything metrical in terms of closed string field theory.
