Inspiron 14-5447, will not charge

I have a Dell Inspiron 14-5447 that no longer can charge. For over a year, I've generally had to charge it while the laptop and the charger had a particular orientation with regard to each other (laptop on end, with charge port down and indicator light up, AC/DC device off to one particular side, DC plug in 95% of the way was my "guaranteed-to-work" orientation). I figured there was something off with the wires or the contacts of my DC cord, requiring a stress to the cable to allow for a connected circuit. My magic voodoo charging method has failed me starting about a week or two ago, rendering the laptop inoperable. Today I tried another Dell charger, (original: LA65NS2-01 19.5V/3.34A; replacement: LA90PM111 19.5V/4.62A; laptop input 19.5V and both 3.34A and 4.62A explicitly listed as compatible on the bottom of my laptop), but this replacement charger doesn't work with this laptop (does work on a Latitude E6420XFR).

I notice when I'm looking at the bottom of the laptop, there are two screwholes in the hinge area, but without screws present.  I have no idea if this has always been the case, or if the screws backed out, dropped off, and got lost. As the charger port is close to one of these screw positions, is it possible that the case isn't being "clamped" enough for good contact in the power port? Can the battery and/or DC power port be replaced on this model?

Serial number sticker on the bottom of my laptop is completely blank (i.e. the ink rubbed off), so I can't seem to get into the dell support system to see if my laptop is under warranty, but I'm guessing it is not.