I plan on spending up to 10k of my own money. At which point should I seek out funding, can I hire someone to run social media and build hype ?
Or should I self publish my small game with what limited resources I have ?
I would suggest to you that you immediately get a few friends to beta test the game and ask for their honest opinion so you can hammer out what you actually have working for you right now before you put any additional money into the project.
Next, if lots of things are broken, re-evaluate if you can fix it and have appetite for that risk.
Then, if you still want to proceed, you have the options of basically pre-funding and post-funding. Pre-funding would involve things like Patreon, selling equity in the project (someone does labor for you in exchange for a share of future profits), and things of that nature. I don't think there are many banks or investors out there who give you money for a project like this, but your friendly 0% interest rate credit card salesman is also an option. There is a great talk about this on GDC's Youtube channel called "How I survived 10 years in game development without a single hit game".
Lastly, if you decide to go with post-funding, your options are basically selling copies of your game on Steam or some app store, and/or charging microtransactions. These things are a little complicated to set up.
Good luck!
I have a nice full time job so I can support myself, I don't think I'd abandon the project if I get bad feedback at this stage. I'd probably just tweak it , if 6k in I still can't make it fun then I'll call it a learning experience