Triple Couloirs might still be in condition in early May, if it comes in at all. One to two days for TC, prolly two for SGC, which should still be in. Other gulleys include North Buttress Couloir and NE Couloir on Colchuck Peak, NE Couloir on Argonaut,and NE Couloir on Dragontail. The Ice Cliff Glacier on Mt Stuart may be of interest to you as well.
In the Icicle Creek valley in Leavenworth are the 8 Mile and Bridge Creek campgrounds, both are Forest Service. I think there are more campgrounds further up the Icicle as well. Additionally, there are some other non official camping options.
Other options include Coleman Headwall and North Ridge of Baker, many options on Rainier, North Face of Shuksan could be good.