If you decide in Albania or Montenegro, be advised that we've been told by the company whose tour we are taking this month not to drink the water in those two countries.
Did the company explain why you shouldn't drink the water? I can't speak for Albania but we were in Montenegro a week ago and drank the water with absolutely no issues whatsoever, in fact it tasted better than some of the water I've drunk in the US. In the old town of Kotor there are public water fountains and people were constantly filling their bottles with the water with seemingly no bad effects. Even if drinking bottled water in a restaurant the ice is made from tap water, none of my family suffered any ill effects from the water in Montenegro.
Whilst on the topic of Montenegro it was listed as a potential place for the OP to visit. I would hesitate to advise it particularly as the OP is looking to visit at peak summer season. Kotor, where we stayed, doesn't have any real beaches to speak of. The shoreline is shingle and rock and lots of pontoons which traditionally were used to secure and access the many fishing boats. Kotor is not cheap, eating out is on a par with most Western European cities.
Some of the best beaches are in Budva however it is a very popular spot, particularly amongst wealthy Russians and probably wouldn't be ideal for the kids. As for safety, Montenegrin drivers are atrocious, I witnessed some very dangerous driving there and some of the roads are hazardous.
The Polish Riviera has interested me and is on my list but I haven't been so cannot make a recommendation.
You don't state the reason for not using a rental car. I would recommend Mallorca but it's best enjoyed by car. Mallorca is a pleasure to drive around, it's not congested (other than Palma) and the driving by Southern Europe standards is particularly mellow. Plenty of beaches, an aquapark, a very good aquarium, castles, caves, walled medieval towns, fantastic markets, reasonably cheap, excellent food and very easy to get around and explore. We've been going for a number of years and our kids love it (now 14 and 12). We've been in June and it's perfect, we've been in August which is hotter but fine if you're at the beach or by the pool or go up into the mountains where temperatures are a bit cooler. It's a very safe place, never experienced any problems or felt unsafe and we sometimes visit twice a year.
Portugal is also highly recommended for value for money, excellent food and a safe place. I've yet to go but we're looking at possibly next year.
Southern Spain will be hot but that doesn't stop the thousands of European tourists who descend upon it every summer. Whilst the heat isn't conducive to walking around and exploring historic sights it doesn't matter much if you're going to be spending much of your time around water.