Correct me if I'm wrong, but looking a the first diagram, what I see
is that when a plane flying at level flight encounters a sideways
gust, it will also make the plane aggressively roll over, just as if
the sideways gust was from a roll induced sideslip.
That's correct. The "positive effective dihedral" generates a "downwind" roll torque in both cases, in relation to the "relative wind".
Is there a way to have my plane react normally to a sideways gust
while having static roll stability?
Just go easy on the dihedral, or on other factors that contribute to "positive effective dihedral" (e.g. high wing placement, low C.G. placement.) "Overdoing" these things, especially in combination, will give you way too much "effective dihedral" which will make your plane much too vulnerable to the immediate effects of strong sideways gusts. And see the comment below about the benefit of reducing the size of the vertical fin.
Do I need way bigger vertical stabilizers for these dihedral values?
I don't believe that that's going to help you. And contrary to another answer, I don't believe that you are experiencing classic "Dutch Roll". I think you are just seeing your plane suffer from the immediate effects of strong gusts, due to excessive "effective dihedral", and likely due to excessive positive longitudinal stability (pitch stability) as well.
For any given desired degree of positive roll stability, you can get away with less "positive effective dihedral" geometry -- specifically meaning that the aircraft will generate less "downwind" roll torque at any given sideslip angle-- if you reduce the size of the vertical fin. This will allow the plane to experience more sideslip in the long run, so that whatever "positive effective dihedral" is present will be more effective at generating a roll torque to roll the plane back to wings-level.1
Look at it this way-- if you want to increase the plane's degree of positive roll stability, you can either increase the degree of "positive effective dihedral", or you can decrease the size of the vertical fin. Either alteration will help the plane come back to wings-level in the long run.
And to a first approximation, we might think that either alteration also might make the plane have a greater tendency to roll "downwind" immediately after being struck by a sudden side-gust. But that's not really true. In the short run, the effects of the two alterations will have very different effects on the way the plane reacts to a sudden side-gust. Because when the plane is struck by a transient side-gust, it doesn't have time to yaw into alignment with the airflow (relative wind) before the "positive effective dihedral" generates a strong "downwind" roll torque, so an overly-large fin won't do much to minimize the aircraft's immediate (downwind) roll response in this situation.
So for best stability in gusty conditions, keep the vertical fin as small as you can, and go easy on the "positive effective dihedral" as well, only designing in the minimum amount needed to slowly return the aircraft to wings-level after a disturbance. The smaller the vertical fin you can get away with, the less "positive effective dihedral" you'll be able to get way with as well, and still have the plane slowly roll back to wings-level after a disturbance.
Generally speaking, all these factors tend to shift the "effective dihedral" in the positive direction:
Low CG placement relative to overall shape of aircraft
Wing mounted directly on top of fuselage, especially a slab-sided fuselage. (To some extent a similar extent is generated by putting the wing on a pylon of substantial length ("chord").)
Actual dihedral in the wing (or tail)
High-mounted vertical fin
Naturally, an important question that would help you choose from the various different solutions that have been offered in various answers, is "how does the plane fly in smooth air?" True "Dutch roll" instability should manifest itself even in smooth air, especially in response to certain initial control inputs. If you do find that to be occurring, then this answer may not be the right one for your situation.
- To understand why "positive effective dihedral" tends to return an aircraft to wings-level after a disturbance, we have to understand that a) all turns tend to involve some amount of sideslip (toward the inside wingtip, so that a "yaw string" would blow toward the outside wingtip), at least in the absence of a "coordinating" rudder input, and b) sideslip interacts with positive effective dihedral"-- no matter exactly what the specific source of the "positive effective dihedral" geometry is -- to create a downwind roll torque, which in the absence of gusts, tends to return an aircraft toward wings-level.