Hey Wendy,
As some have alluded to the tips of the bullets are usually not overly consistent. You can easily check this by simply measuring some of them from tip to bases. Even with some of the plastic tipped ones like Hornady the ogives will vary a smidge in profile from one to the next. You have to use a
comparator to check this. If you find there is a significant issue this may solve some of your problem, since most of the seating dies have the hollow stem which sits down on the ogive a bit. In some cases however, depending on the length or profile of the ogive, the tip may still hit the bottom of the seating stem. In this case you might have to have it relieved a touch or do so yourself if you have a good drip press and are confident using it.
As to the issue your having with your press, yes this can induce the differences your seeing real easily. I have several brands and sizes of of presses, Lee, Wells, RCBS, and I use the heaviest cast versions for my more accurate loads. It doesn't move any what so ever when I put normal pressure on it when sizing or seating especially. In fact you shouldn't have much pressure when seating anyway, so if your seeing things move when doing this step you might consider picking up a solid cast steel press for your top end loading.
The only other thing I can add to this would be to set your length, load 10 rounds, 5 seated normally and 5 seated meticulously as you mentioned above and shoot them against each other comparing the results. If there is a decided notable difference then purchase the comparator set and separate the bullets by length before loading and or pick up a heavier duty cast press.
I understand this is an annoying issue to say the least, however it isn't the end of the world. Take a deep breath and use it as a learning experience and work through it as such. There are a ton of other issues in life that can be twice as annoying and aggravating.
Hope this helps.