Winging it in Ireland is quite easy - and adventurous - even in the summer months, when you can roll into any city, town or village and find a room by knocking on a door or two, or calling ahead that morning. We've done so on numerous, multi-week bicycle trips all over the island and had difficulty finding a room right away on but one occasion when we arrived in a popular commerical town (Westport, Co. Mayo) on a weekend. It took less than an hour to do so eventually, however. If flexibility is what you're looking for, then by all means go for it. And in late September and October, the tourist traffic is barely noticable. If you go to County Donegal, then tourist traffic is basically nil any time of year.
Irish Tourist Board/ITB-approved B&Bs (of which there are over 2,000) are also a lot less expensive than the $80 Airbnb price you cite. Generally expect to pay $45-$65 a night per room. Split that in two and voila! You'll also get a delicious, filling breakfast ("full Irish" or cold), meet the family and make new friends. I stayed at the same B&B at the beginning and end of six of my nine trips to Ireland and watched the four kids grow up over the years. Quite a treat.
So when you arrive in town, just look for an ITB sign with the green shamrock out front. As one of my students says, "Easy-peasy, lemon-squeezy!" B&Bs are everywhere, the Irish exceptionally friendly and drivers patient and courteous.
Here are two B&B websites you can research:
With only ten days in-country and trying to see all you can, you'll be in the car all day long and end up actually seeing (and experiencing) very little. A wiser strategy is to pick a town or two or three in different regions and take day trips from there. Alternatively - and perhaps the optimum way to see and experience a good, managable slice of Ireland, its people and culture (e.g., historic sites, castles, pub life, trad music and such) - is to focus on a couple of areas or two or three or four counties and pick and choose your routes and overnight spots each day. That's flexibility.
But first, answer these for yourself: What are your interests? What do you want to see and do? Do you like rugged coastlines, the mountains, villages, small towns or cities (of which there are only three or four), beaches, history, great scenery? Once you've determined that, then the structure and objectives of your itinerary will fall into place much easier.