I've only tried GMless with another player once, and I feel like I took over the game, even though I didn't plan to.
Having a dominant player control the game's flow is definitely going to be an issue, whether they try to push their story or not. On the other hand... that's an issue in regular game too, the only difference is that the GM can try to draw out the other players normally.
The best way around that would be along the lines of Nagatobimaru's suggestion: making post that other players can continue on. The keywords he suggest are one option, another would be to let others answer your questions. As in, you stop your post at the point where you would ask the GM what happens next.
For example: your group arrives at a door. You want to know what's behind it, but only after checking for traps, and if you can unlock it. So you make the relevant rolls to search for a trap, and ask what happens. The next player to post decides if there is a trap (by deciding how likely it is and rolling randomly, most of the time), then uses the game's guideline to see if you succeed in finding it. He then either describe the trap you find (or trigger, if you didn't spot it), then if you can pass, he asks what's beyond the door. The next player picks up the ball, and describe the room, and so on.
Overpowering PC isn't too likely an issue, I think. Sure, you'll be tempted to give yourself cool stuff, but at the same time you'll feel worried you're giving yourself too much. How to deal with that depends on the rules you want to use. Pathfinder, 3.5 and 4e had general reward tables based on character level.
5e on the other hand... seems pretty vague as far as I could tell. Except for Appendix A of Xanathar's Guide to Everything. It suggest giving player a fix amount of gold every time they level up, and a certain number of Item Points for them to purchase magical items on a list of approved rewards. It may take away from the dungeon looting experience, but since you want to focus more on character interaction, that could be a good work-around.
The biggest hurdle will be the direction and format. When you play solo, that's not an issue: you're the only one reading the prompt from the GME, so whatever happens will be what makes the most sense to you. If you're convinced your country's invaded, it won't turn into Mars Attack unless you want it to. In a group, there's no real way to insure the other players won't throw little green man in.
The only solution I can think of would be to push for a flavor that fits the rules. That means encouraging players to use the Monster Manual for opponents, sticking to classical medieval fantasy tropes, and so forth.
To look into GME, I'd recommend looking at the following:
http://www.evilgenius.org.nz/blackmagic.html
A simple HTML version of Mythic. Mostly used to answer Yes/No questions, with the occasional Random Event thrown in. It's pretty basic, but workable. Just keep in mind that there is a bug where rolling exactly what is needed for a Yes gives an Ex No answer instead.
https://www.rpgsolo.com/
A more complex GME, giving Yes/No answers, using occasional And/But rather than Ex answers. Doesn't give out Random Event on the usual questions, but it includes Mythic's normal Event table. It also includes a few other generators, including UNE for quick NPCs.
https://thecollaborativegamer.wordpress.com/worlds/
Although it's not really a GME, the way it handles adventures is a must. Each dungeons (and world regions) has three main monster type that you are more likely to encounter, allowing for variety without turning into a full free-for-all. The actual exploration is slimmed down and abstracted, dealing more with 'what is in this part of the dungeon' than how many doors and empty rooms are in each corridors. And the use of Time Unit to track how long the adventure takes is a good way to avoid the "10 minute day" of older editions.
