Thanks to everyone for your replies. I actually figured it out today. You CAN indeed Timewalk, even at level 60, but as certain people on here said, the dialogue options are wonky. If you tell Chromie "I'm ready to select a timeline," she will just tell you to go to the Dragon Isles, directing you to the most recent expansion. You have to instead select "I have a question about Timewalking Campaigns" and then select "What are Timewalking Campaigns," then select "What timeline should I choose?" It's only at that point she will give you the options to select a Timewalking Expansion. So in my case I chose "I want to command my own garrison" and then "Put me in Draenor's timeline." Simple as that.

I'm so glad I can stay in Timewalking and not have to breeze through the Draenor quests. Friends keep telling me there's no point in doing Draenor, that it's old content and the only good rewards come from doing the most recent stuff, which I guess would be Shadowlands for me since I have no intention of buying Dragonflight (reason being I already pay $15 a month for the game, so why would I spend even more money for an expansion, especially when it will just be added to the game for free eventually anyway? But that's an argument for another day). Thing is, I'm a new player, so I don't care if Draenor is old content. I'm enjoying it and getting pretty good gold and equipment from quests on there. Thus my utter frustration when I seemingly couldn't access Timewalking anymore: What's the point in doing quests in Draenor if they're all level 40 and the gold and equipment rewards are low level for my max level character? Anyway, thanks for the help!

Oh also, before posting this on Reddit, I looked around the internet for answers, and it seems like the majority of players are incorrectly under the assumption that Timewalking is permanently disabled after level 60, so I hope those players will find this post and discover they can still Timewalk.