We were in eastern med in July 2023 during the heatwave, 7-day cruise though not with
DCL. Every port day was 105+. Greece (which is where we had multiple stops and ended the cruise) was even hotter. Acropolis was 115-117 that week. For us that wasn't a huge burden but it does slow you down and for us it presented a problem in Athens when sites would close for the bulk of the day due to the heat. However, many of the tour guides were clearly uncomfortable. Drink plenty of water too. The cruise line we go with provides water for you and just had a table of water bottles as you got off the ship for the excursions but I know DCL is different (I think you have to buy the bottled water package?). Anyways yes plan for dehydration concerns.
We just did a western med march/april 2025 much better weather although sometimes it was warmer than was anticipated, monte carlo (where we ended) felt warmer than it should so I was hot in my long-sleeve athletic top.
We're headed to Switzerland in July 2026. I didn't want to go during then but that's how it worked with airfare. It won't be nearly as hot as the Med though.
Personally the crowds in July 2023 weren't terrible but we were in Venice, Croatia, Montenegro and Greece so less popular in terms of crowds. March/April 2025 crowds were obviously lower as it's the off season/very beginning of the season.
Back to your cruise: One of my main recommendations is honestly get a sun hat. I don't care if you think you'll look dumb in it like my husband doesn't really like to wear it but just get it and wear it. It helped me a lot in terms of having a slight cooling effect and some protection from the sun
This was us at an ancient greek academy on that July 2023 cruise for example. I have two of them. The one pictured has a slot in the back for you to put your hair through if you wear a ponytail...or if you don't (like I didn't) it still works. The other is a tan one that is very similar in style but doesn't have the slot in the back for your hair.
Obviously bring sunscreen and bring it with you while you're out on excursions to reapply.