If you absolutely, positively need to run a specific Dungeon as a F2P player, simply run to the portal and enter manually, your whole group will get ported in, you'll be hard pressed to find people who will run a specific Dungeon because you can only get the tokens from Random queue with charges, obviously Warfronts don't have portals so you're stuck with Random queue only as F2P
At the end of the day the essentials "must have" unlocks are Auction House (€/$5) and earring slots, maybe Planewalker: Water and...that's it, if you simply, absolutely must have everything for whatever reason like extra souls, new calling then pony up and buy, no such thing as a free lunch, money has to come from somewhere but bear in mind, even as a F2P player you are not restricted to the content and how often you can run said content.
If you want to run Dungeons, Warfronts, Instant Adventures 24/7 until your eyes bleed, you can, with no restrictions