I just returned from a week of diving in Cozumel, so I can attest to what it was like in late June.
Many of the dives were "drift dives" or "live boat" dives, and most had noticeable currents, but not all of them. Around Palancar we did several dives (Palancar Caves and Palancar Horseshoe in particular) that had very little current evident. We went through, under, around all sorts of formations and it was wonderful.
We also did one "freeway dive" that I call the "Gran Prix de Cozumel". We started at Punta Dalila Reef and blew right past that and La Francesa and Paso Del Cedral reefs. The DM estimated the current at 5 knots. (might be exagerated slightly, but it was Ripping!) The current was too strong to fight, and to fast to really get a chance to see much until we started our ascents.
The DM's talked about the "weird currents" we were finding. Aparently some of the currents change significantly in May~June, but they usually settle back out by late June, so they were surprised that they were so unpredictable on the 20th~21st.
Our DMs used a sausage during the SS as described, and the boat always picked us up right away, except for one time when we got broke into two separate groups. (We were checking some big Cudas and the girls drifted beyond us and encountered a turtle the "size of a dining table")
I guess the "rule" would be to expect currents on the walls and open reef dives, but in some of the more protected areas you can really relax and enjoy the scenery up close and personal. Stay close to the reef (but not too close) because the current is usually gentler closer to the reef, and just make sure that you stay with the group.
Wristshot: