Since rocks can have magnetic signatures, as is the case with paleomagnetism, can they store past pressures also?
Yes, but not directly, and it is very hard to know the exact pressure (in contrast with paleomagnetism where you can actually get a nice number).
One way to do it is to simply look at the minerals you have in the rock. Some minerals exist only in a certain range of pressures. The best and most known example is the two polymorphs of carbon: graphite and diamond. If you find a rock with diamonds in it then you know that (at least the diamonds) formed at great depth. There are many other mineral transformations that depend on pressure. Andalusite is low pressure whereas kyanite is high pressure. Plagioclase is low pressure, spinel is intermediate pressure and garnet is high pressure. You can look up images of those minerals for some eye candy! Note that I'm using the words "low" and "high" quite vaguely, as everything is relative and the exact numbers are not important for this case.
Another way of finding the pressure of a rock formation is by using the chemical compositions of the minerals in it. This is basically an improved version of the previous method. Instead of having a certain mineral assemblage transform to a second assemblage, we take two coexisting minerals. If we know how a certain element (for example magnesium or aluminum or whatever) behaves in a certain fashion relative to the pressure, we can measure that element in both minerals and find out the pressure.
A third way is to look at a thing called fluid inclusions. These are tiny bubbles of liquid or gas (or something in between, the distinction may be fuzzy at high pressures and temperatures), trapped in a mineral when it was forming. This is essentially a tiny pressure vessel, so using all kinds of methods we can figure out what was the pressure during that time. Here's a picture of how it looks under the microscope:
But is there any way they can look at rock morphology to see how high sea levels may have been?
Not using these methods. The problem is sensitivity. I was talking to you about diamond and graphite - this is extremely deep in the Earth. The pressure of adding a few metres of ocean on top of it are definitely not even going to tickle that transformation. Furthermore, for this to actually be effective, the rocks have to be directly on the seafloor. Once you start burying them in the sediment, you have additional pressure on top of them. A general rule of thumb is 100 metres of additional rock pressure equal 300 metres of water pressure. How would you know how much rock was on top of it? And as the rock gets buried deeper, how would you know what time frame your measurement represents?
So in short, no. We can't do it. There are, however, other methods of figuring out sea levels without using pressures.