I just bought my first electric having been playing acoustic for a year or so

I'm finding that the tune of a note depends A LOT on how hard I press down my fingers. This was not the case on my acoustic, but I'm noticing a marked difference in pitch, depending on whether I press soft or hard. this makes a well tuned guitar then seem just that horrible little bit out of tune...

Is this normal? Anything I can do about it?
It is perfectly normal. It's something you have to get used to when playing electric guitar. You'll get used to it, you just have to find the right amount of pressure to apply to the string. Don't press down too hard or you will be sharp.

Part of the 7-string Legion

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and my 7-string Ernie Ball MM JP-7 build

yeah theres something you can do about it...be happy..lol

You have just been introduced to the learning curve that comes
from learning on acoustic first. Your ear is trained and developing
very nicely. Now you have an understanding on the dynamics of
the neck. You may not even need as many pedals now..
I bet Charlie Brown's teacher's name was Mrs.Hammett