Exposure wise the difference between Aperture Priority with exposure adjustment and fully manual Is one of preference rather than result. you end up with the same settings. generally speaking it is quicker to use the former.
Shooting in raw with individual processing is the way to go for consistency.
Grave yard and grave stones are not the easiest things to photograph. and for many reasons.
A vast majority of Christian burials face East and the rising sun. so there is only a narrow time of day when the sun will be falling obliquely across the face of the stone to give maximum texture and detail. however one can choose to back light the stones as you have done, but this gives a very flat light on the stone itself. and for older stones will probably result in them being difficult to read.
many stone have vertical sides and this needs to be maintained in the image. This can be done either by careful positioning of the camera or later in software. which has the major advantage of being able to choose a wider range of viewpoints. The loss in quality, when using a 24megapixel camera like yours, is now pretty insignificant when making such adjustments. A costly alternative is is to use a tilt shift lens, which I think would not be justified for this work.
in terms of speed and convenience A mirrorless camera is far easier to use in these situations especially when using live view, But you have what you have, and the final result is unlikely to be any different.
I would not aim for an Identical look but rather the best that you can do in each situation.
The three images that you show show a similar full tonal range and general brightness level, which is good. however There are other occasions when such a look would not be appropriate, such as a grave stone in soft morning light through a light mist. or a grave stone at night lit by a shaft of street light.
Variety is generally more acceptable than sameness. the handling that you choose should reflect the location, the situation and the wishes of the family. However even if they only want the simplest record always add you own best interpretation as an alternative. ( this is true of almost any photography)
As to Comment on the three images that you show. The first two are very acceptable for shots of black grave stones in shadow.
I am not convinced that they are exactly vertical and the stones look nice and clean. however you did not check the surround very well and you should have removed the sticks and lumps included in the frame. such attention to detail are important. I would consider using off camera Fill flash in some circumstances to liven the images.
The shot of the door is a situation where the viewpoint and framing could be better. you have clearly chosen the height to minimise keystoning.
In this instance it would have been better to give priority to the framing of the door with the branches, and also reducing the emphasis of the drive way . Then making corrections to the vertical in post processing. again there are a few leaves gum on the path that would have been better removed. as the catch the eye.
That is a question of learning what to tidy up and what to leave to be cleaned up in PP. I do like the general concept and the lighting.