First of all, let's temporarily make a distinction between ionic and covalent compounds. Generally speaking, covalent compounds are more soluble in nonpolar solvents while ionic compounds are more soluble in polar solvents. Of course ionic and covalent bonding exist on a spectrum and some compounds you would expect to be ionic are indeed more covalent in nature. This concept is often visualized using the Van-Arkel Ketelaar triangle of bonding
As described by the Fajans' Rules, highly polarizing cations will distort polarizable anions, pulling their electrons into the inter-atomic space to the extent that they essentially form ionic compounds. This is one of the reasons solubility decreases with increased polarity. Solubility does not increase with increased covalent nature. In water, it would decrease. The best example of this is beryllium. Almost all $\ce{Be^{2+}}$ compounds actually exhibit covalent chemistry, as we can see with beryllium peroxide in the Van-Arkel Ketelaar triangle above. This is because beryllium has extremely high charge density.
In order to properly understand solubility trends of ionic compounds, we should examine the enthalpy of dissolution using a Born-Haber cycle.
The first step is vaporization of the crystalline lattice, which has a large positive enthalpy. This is affected by a few factors, but it basically represents the stability of the crystal. Generally, dipositive anions/cations will have much greater lattice energy because of stronger electrostatic interactions, and lattice match should also be considered. A significant mismatch in size will lead to lower lattice energy. In the Born-Haber cycle, the ion-ion interactions are broken when the crystal is vaporized, then ion-dipole interactions are formed when water molecules surround the gaseous ions, so the other factor to consider in the Born-Haber cycle of dissolution is hydration enthalpy, which is negative. Generally, this is increased with increasing charge density. Sometimes net enthalpy is positive, and in these cases, entropy may also come into play. For the most part, you can think of ionic solubility as a balance between lattice energy and hydration enthalpy.
Solubility increases with decreased lattice enthalpy increased hydration enthalpy
and you're right that
Solubility decreases with increased polarizability
Sodium's ionic radius is 227 pm, while ammonium's is 175. Often ammonium is a good substitute for sodium, as a large, low charge density cation (for stabilizing highly polarizable anions like bicarbonate). In fact, ammonium bicarbonate is one of the only soluble non-group 1 bicarbonates. Still, ammonium is still a little smaller than sodium. As a solid, ammonium will stabilize chloride better, but bicarbonate worse.
Since bicarbonate is much much more polarizable than chlorine, the two bicarbonate salts will probably be less soluble than the two chlorides. Ammonium will stabilize chloride better by size match, so ammonium chloride will be a little less soluble than sodium chloride. Due to it's large size and low CD, though, sodium will stabilize bicarbonate better, so sodium bicarbonate will be a little less soluble than ammonium carbonate.
The way I see it, the order should be:
$\ce{NaCl}$ > $\ce{NH4Cl}$ > $\ce{NH4HCO3}$ > $\ce{NaHCO3}$ (decreasing solubility)
Of course solubility is a pretty complex concept and these are only generalizations. To get a better handle on it, I would try looking at the enthalpy and entropy cycles of different types of salts in each of the main groups. Trends exist, but the way they work is a little different depending on the group. I think, for example, that lattice match is mostly important in group 1 salts because of their lattice types. Anion charge becomes an important factor in group 2. Trends can be established but a lot of the time they are only consistent within a given group.
