Overheating 2003 Honda civic?

2003 Civic Ex Coupe with an automatic transmission. A couple months ago the heater core went out, leaking antifreeze into the passenger foot area, but I couldn't afford a mechanic, so for the time being I cut the hoses going to and from the heater core and coupled them together (I'm not sure if this has any impact on the current issue or not). After doing so I filled up both the radiator and the reservoir. Recently I've noticed that the engine temperature gets pretty hot in slow or stop and go traffic, never getting into the red. However today, in stop and go, it made it past the red, I pulled over, the smoke or steam was coming directly from the radiator cap, there was splashed antifreeze around the cap as if it was boiling out of the cap. However when I went to add water to the reservoir, it was totally full, exactly how it was a few months ago. I waited until the engine was cooler and continued to drive to work, however the engine didn't get cooler as I reached higher speeds like it used to it would just heat up and stay hot. So I pulled over, repeated the process until I got to work. I did notice both radiator fans were running, as I thought it was a fan issue
Based off of what I've searched it seems some of the possible issues include air in the radiator, bad radiator and/or thermostat, blown head gasket, or the radiator is out of coolant
Any ideas as to, What the problem is? How to fix the problem? How to get the car home in one piece to fix it without getting a tow?

Although it could be few issues as you have mentioned, I would be leaning more towards that you may have a blown cylinder head gasket. When your cylinder head gasket fails it allows combustion pressure to enter the cooling passages of the engine block. This increase in pressure may have been the reason your heater core had failed as it may have not been able to hold the increased pressure and either split an end tank, fitting or even perforated a week piece of the heater core itself. With increased rpm while driving the pressure builds up in the cooling system and forces the coolant away from the engine block and into the radiator. It also stops the circulation of coolant and will cause your car to overheat very quickly while driving and in some cases when stopped and without the increase in pressure your car temperature may fall as the coolant can now circulate through the cooling system. On a cold engine top up the radiator and with the cap off start the engine. Continued below…

You have to bite the bullet and get a new radiator. Yours is rusted inside.