If I Switch my Active Pokemon that is affected by Poison from Muk & Alolan Muk GX's "Severe Poison" attack and Dust Island is in play, does my new Active Pokemon take the extra damage counters?
Yes, it does. Dust Island specifies that it is affected by "that Special Condition".
Source: Unbroken Bonds FAQ (2019-04-19), TPCi Rules Team (2019-05-02)

Relevant cards

Note: All applicable cards may not be listed below.
SM10_196
SM10_196
SM10_197
SM10_197
SM10_220
SM10_220
SM10_61
SM10_61