Not knowing what you like and with the obvious caveat we all are different here are some thoughts. First the biggie....you are so right trying to cram in as much as possible is the exact wrong way to go. More than any other place we've been Ireland is best enjoyed leisurely ....less is more. That doesn't mean not to see a bunch of stuff...it means not pushing all the time, avoiding one night stays when possible and enjoy the many riches of a given area while not being in a car too much.
Honestly can't imagine being in Dublin for 9-10 days....it's okay but it's a city....a fairly modern city. There are nice day trips from Dublin.....Trim, Newgrange, Tara to the north.......or the gorgeous Wicklow mountains and Glendalough to the south. Also a day trip to Belfast is also well worth it.
But I would definitely bite that bullet and rent a car and head out for 3-4 days. Cork is another big city without as many attractions as Dublin. Places near it like Kinsale are nice but it would be low on my list. In Ireland....pick a spot, almost any spot, particularly in the west and there are many days/weeks of things to do very close by.
But here's one option we LOVED. We stayed in the Castlemaine/Milltown area (though Glenbeigh, Kilorglin and Castlemaine would do) This is a wonderful small Irish town with a few great pubs, restaurants and shops nearby that give you a real taste of wild Ireland. But what is best is that you would be perfectly situated for both Dingle and the Ring of Kerry and all the amazing sights near (not in) Killarney. You would be within an 1- 1 1/2 hours at most of all of Dingle, the Skellig Ring and Killarney sights......often much closer. So just down the road you would have on Dingle....Inch Beach, Beehive huts, old forts and ruins, Dingle Town, the harbor with dolphin cruises or cruises to Great Blasket Island (incredible), Slea Head Drive (again incredible), the Gallarus Oratory, Dunquin, Connor Pass and fantastic, shops, restaurants, pubs and music. Half hour away near Killarney you would have the Gap of Dunloe, Ross Castle, Muckross House, Torc Waterfall, Ladies View, the National Park, Kissane Sheep Farm and lots more.... Then in the ROK you would have insane scenery (our favorite stretch above Caherdaniel), Great small towns like Kenmare, Sneem, Waterville, Cahersiveen, Glenbeigh and more with ancient ring forts, castle ruins all against the backdrop of the Magillicuddy Reeks. Then there is the Skellig Ring with the insanely beautiful cliffs of Kerry (far less touristy than the Cliffs of Moher and IMO prettier too), Skellig Michael, Bray Head, Valencia Island, McCarthys Castle, Abby Ruins, Skelligs chocolates, the Glen, the beach at Ballinskellig...all on a road with no tour buses allowed. You literally could spend weeks/months there and not exhaust things to do. We have and there's plenty left. So in 3-4 days you could have your choice of a myriad of options....and at that time of year there are no worries about it being tourist clogged.
Best of all .... the Irish folks in Kerry are the warmest, down to earth, funniest people we've ever come across. Often they'll tease you at first, see how you react and if you're not full of yourself they treat you like family.
Anyway....that's one great alternative. There are many. Check out YouTube videos of the sights and see what suits you. But keep that great philosophy of not cramming in too much and try to see a spot well away from Dublin. It's worth it. Have a great trip.