@Laura4711 First of all, it sounds like you are submitting Booking Requests, correct? As opposed to Inquiries?
Either one is fine to use- if you have any questions for the host before wanting to commit to a booking, an Inquiry, ( the Contact Host button) is appropriate. The host then has the choice to pre-approve, (in which case you can then go on to book), or decline, or just answer your questions. They don't have to pre-approve or decline, just answer, so many, if not most hosts will just answer. If you get the answers you are looking for that lead you to want to book and the Inquiry has expired, just go on to submit a booking request.
For a booking request, just the message you send with the request is all that's expected. Hosts like to receive requests where the guest has provided info about their group, made it clear that they have thoroughly read through all the listing info, including the house rules, and maybe give a little indication of the reason for booking (coming to attend a wedding, visit museums, go hiking, etc), and have entered the correct number of guests on the booking.
If the host has questions they'd like answered, they may just message you back before deciding to accept, based on your answers and how promptly you reply. A host must either accept or decline a request within 24 hours or take a hit on their response rate.
As far as a price going up, some hosts may do this unethically, because they've had a lot of requests for those dates, and realize they could charge more, sometimes it's just a matter of them not having remembered to update their prices for a holiday or major event period when they can charge more.
A couple of things that may be affecting your attempts. Many hosts are reluctant to accept bookings from guests with no prior reviews. Of course this makes things more difficult for guests new to the platform- someone has to take a chance on you in order for you to get your first booking and review. Many hosts have had terrible experiences with unreviewed guests who leave a mess behind them, sneak in extra people or pets, throw a party and trash the house and aren't willing to take a chance again. I'm sure your family isn't in those categories of bad guests, but hosts may just not accept new to Airbnb guests on principle.
The other thing is that perhaps you have been sending requests to hosts who normally use Instant Book. You can't IB at this point, because those hosts likely have an IB requirement that guests must have gotten a thumbs up from previous hosts, so your attempt to book their place comes through to them as a request from someone who doesn't meet their IB criteria. That may make them wary, even though you've done nothing wrong.
So you might have better luck looking at listings where the host doesn't use the Instant Book option at all and requires all guests to send booking requests. Those hosts are much more accustomed to dialoguing with guests before accepting, answering request and inquiry messages promptly, and have their own criteria for feeling out whether a guest is a good bet to accept, rather than using the standard IB criteria.
I can't see you are doing anything wrong, you just haven't been lucky in selecting listings where the host will respond promptly and professionally and maybe just doesn't want to accept newbie guests.
Also make sure to read the reviews to see if there is any mention of the host not communicating well or promptly.
Keep trying, I'm sure you'll find something.