A few different things are going on here, but rest assured you can carbonate beer in a much shorter timeframe than the rule of thumb for bottle conditioning (3 weeks at 70°F, and longer if its cooler and for high abv and sour beers).
First of all, it sounds like you were using the "burst carbonation method".
It helps to understand the relationship between solubility of CO2 and temperature: simply CO2 dissolves better in cold beer, and poorly in warm beer.
In your case, you tried to rapidly carbonate warm beer. It might have worked if you had chilled the beer fully to 38°F before putting the gas on.
Then when you found flat beer, you used the "rock and roll method" aka "crank and shake method" on very cold beer. One of the drawbacks to both the burst carbonation method and especially crank and shake is that it is very easy to overcarbonate the beer. That's probably what happened to your beer.
The other common method is "set it and forget it": set the psi for exactly the amount that the Zahm & Nagel chart for force carbonation of beer indicates for your beer's temperature.
The method for burst carbonation is to start with cold beer, crank the psi up to a very high number (30-45 psi is common), wait a certain amount of time (8-24 hours is common), and then gently vent the head space, reset the pressure to the psi indicated for your carbonation level and beer temp, and wait another 6-12 hours for everything to equilibrate. The devil is in the details when it comes to burst carbonation, and you sort of have to figure out what works for you, being conservative rather than aggressive if you don't want to overcarbonate your beer.
In terms of solving your current problem, which is likely overcarbonation, remove the beer from the fridge and warm it up and meanwhile vent the headspace every few hours, until the carbonation level is reduced. Then start over using one of those methods, remembering that you're probably not starting from "uncarbonated" beer any more.
Another possibility that is mentioned is that your beer lines may be too warm -- especially if you have a draft tower and the second or third pint is pouring fine once the line is cooled down -- or too short. The beer line offers a certain amount of resistance per length (per foot), and if you for example have the regulator set to 12 psi, you are counting on dropping the pressure just enough so beer is barely falling out (instead of shooting out and foaming) due to a combination of gravity (pushing beer up to the height of the faucet) and resistance from how ever many feet of draft tubing you have. Many people find it makes sense to replace their beer line with 10-12' of tubing, and then cut it back as needed to "balance" their draft system. Others do some reading online and find the line balancing equation and do it mathematically, rather than by trial and error.