Carbonation in keg

You don't say that you chilled the keg before beginning the carbonation process. This is a critical step.

The solubility of CO2 is quite low at room/fermentation temperatures. Before you can successfully force-carbonate with high pressure and shaking, you have to cool the keg. In my experience, you need to plan on at least 24 hours in the fridge before you can high-pressure carbonate.

Over the years, I have transitioned from high pressure and shaking to a week at more modest pressure (e.g., 10 PSI) for a couple of reasons:

1. High pressure with shaking gave me really inconsistent results. Either I would under-carbonate and have to re-pressurize and shake some more, or I would over-carbonate and have to deal with excessive foaming (especially with short serving lines).
2. High pressure with shaking caused some marital difficulties, because I would invariably get impatient and spray beer/foam out of the keg at 30+ PSI when de-pressurizing. Apparently, I am not very good at finding and wiping up all of the sticky mess.
3. Modest pressure for a week takes a lot less of my time and reliably gives consistent results. When I keg, I put it in the fridge and connect the CO2 and leave it alone for a week. Versus having to visit the keg a day after kegging (once it is properly chilled) to pressurize, shake, let it settle down for a while, purge the excess pressure, try to clean up the mess I made, test the carbonation level, repeat to fix the carbonation level, etc.

I have also learned that having proper (diameter and length) serving lines is a huge help to make serving not be a frustrating endeavor.

I’ve learned that the best idea is to put yourself through the extra 5-7 days and carbonate it on serving pressure. Thanks for the help.

There are several reasons it could be foaming.

You say you let it sit for an hour at 35 PSI then changed it to 10-12 PSI. If you didn't purge the keg of CO2 between that, then what was inside the keg was still 35 PSI and thus the beer was carbed to that level. Hence, foam.

The length of beer line can affect foaming. A minimum length of 5m (16ft) is suggested for beer lines to prevent foaming.

The temperature of the kegerator/beer will also be a large factor. See this table for help on that.

When pouring the beer make sure you open the tap fully and quickly. Holding it half open will cause lots of foaming to happen.

It might take a little bit longer but the slow forced carbonation method of letting it just sit at serving pressure for 5-7 days is far better and less prone to error.

Also note that the tap plays a large role in foam production. A proper beer tap has a conical channel, so as to gradually decrease the pressure under which the beer finds itself, and no sharp edges that could turbulate the beer and provide nucleation sites. Using inexpensive plastic taps or valves intended for fruit juices and the like will give you nothing but foam no matter what you do.

I'd recommend searching more widely for this information. This is a pretty common issue for first time keggers, and the solutions are widely published. The usual advice is to not do the shake method, especially if you're new to the kegging game, but instead to fill, purge, then leave at serving pressure for 5-10 days. This is a foolproof method. Once you're happy that your system functions properly, you can experiment with other carbonation methods.