Family hotels

Family rooms in Europe tend to have one double bed and two singles. You can search on booking.com and filter for the number of people. Once you find something you like, you can book directly with the hotels.

I did not check recently, but Hotel Messner in Venice used to have reasonably priced family rooms. The Dorsoduro location is good. We did not end up staying there so cannot actually vouch for them.
Venice is always pricy. Think of the complicated logistics of getting all the necessities delivered by small boat. Go out in the early morning and watch the deliveries and you will understand. It is complicated and labor-intensive.

Anyplace in Venice is safe and clean. They may or may not have AC but you should be able to filter for that when you search on booking.com. Note that AC in Italy may not be as strong and effective as what you may find in the US.

Finding it hard to find resonable priced hotels that have two double
beds in one room.

You are in Europe. Don't expect everything to be just like "good ole USA". A lot of European accommodations are for two people, in a double bed or two singles. Europeans travel differently than we do. Larger groups, families, often stay in vacation apartments (Ferienwohnungen, in German) where they have multiple bedrooms and a kitchen.

Places listed in Venice were rather pricey

Yeah, you've selected one of the pricier places in Europe. Try booking a room in NYC.

Are there more options that weren't published

Probably not in Venice. I don't try to be an expert in Italy, but I do know Germany. There, if you find a smaller town's website, you will find 2 to 3 times as many places as you will find in guidebooks or booking websites, and they will be more economical.

With less than a month to go, its highly probable the nice, reasonably priced hotels are already booked.

Finding something could be difficult. Good luck

Try AirBnB or homeaway.

Alas, you've hit a "perfect storm" of issues:

You're booking close to your stay, when many of the best deals will have already been snapped up.
You're looking for peak season in popular places - many smaller accommodations will be sold out.
You're looking for a larger room by European standards (as Lee said, many rooms only accommodate two people).
You're looking in at least one expensive city (Venice) and one moderately expensive city (Paris). You didn't say what other cities in Italy you're looking for.
You want AC (not a standard item in Europe).

At this point, look at the following websites, and accept that you may have to compromise on something (price, AC, location, etc):

https://www.booking.com/
https://www.hotels.com/
https://www.hrs.de/web3/changeLanguage.do;jsessionid=9FC62730B6870DE8B8C521F625C87FBD.46-3?clientId=ZGVfX05FWFQ-&cid=46-3

This last (HRS) is based in Europe, so they may show different rooms from the others (not sure).

Good luck!

We have booked rooms in Venice for several people at a time and we always had to up our price for Venice. We've had multi bedded rooms at the Messner, which Lola mentions, and our son and his wife have stayed there. Everyone has liked the location. As of a few years ago I know they didn't have air conditioning so bring mosquito repellent if that is still the case. Some of the rooms for 4 had a double bed and two singles (which could be pushed together). Count on Venice being a city where you may have to increase your budget without getting luxury.

There is also a possibility that some hotels may have access to an apartment not listed on their website. We had students stay at the Mecure a few years ago but we didn't all fit so they put some in a very nearby apartment. It never hurts to email the hotel directly and ask about a family room for 4. Air conditioning may be a little harder to find though.

Also keep in mind that most European hotels charge per person and not per room. You cannot book a room with two beds for two people and put four people in the room and that includes children.

Also keep in mind that most European hotels charge per person and not
per room

I've seen it both ways. Sometimes it is just a uniform charge per person, regardless of whether it's one person in a single or two persons in a double. More often it is per person, but it is less per person if there are two persons in a double and more if it is one person in a double.

A big reason to charge per person is that many places include breakfast, and that is a major part of the cost of providing the room.

A place I've stayed in in Berchtesgaden only has double rooms and they charge by the room, with a higher charge depending on the number of people.. They have one charge for a double room for 2 people and another, more than half the 2 people charge, for one person in a double. (This place also has a family room that can accommodate 4 people, but the price is for the room, not per person.)