PLEASE help me get on with my MIL!

Gosh stroking hair? How odd!

If I were you I would reply to her frequent contacts only once a day/ 2 days and always late at night or v early.
e.g. "Now I have a moment to say hi! So busy with a baby and tiring. Am off to sleep now, Will let you know if we have time to catch up with you at the week end/next month/tomorrow!"

I would ask her to look after baby for an hour or so each week/fortnight (or whatever suits you) so you can catch up with a friend for coffee/ have your nails done/browse the shops /go for a walk...

If this woman is really like this then you need to set the parameters of what you can manage and that will not make her feel excluded. She may calm down when she sees she has a place in your life!

Was she like this before the baby arrived? Is there a FIL? If its recent I wonder if her emotions have got all over the place with the babes arrival. This is very difficult for you and for the sake of the future I think you will have to cut back visits and make it quite clear you don't like being stroked etc. As far as the texts go answer the first one of the day to confirm you are alright, otherwise you may find her on the doorstep, and then just leave it unless you wish to communicate on that day.

It's practically impossible to change any adults behaviour but what you can change is your reaction to their behaviour.
So, you've got a bit of a dappy Mil but all she is doing is trying, in her own way to be nice to you.
Ok! It's not coming across very well and she's not exactly your cup of tea but she's pretty harmless.
If the texts and messages irritate you, don't look at them, perhaps go through them all, once a day, while rolling your eyes and then give her one big "thankyou" text/message,saying you've been busy.
You say that you feel she is trying to compete for your time and affection with your own Dm but if that is really what she's doing, she's not going to win that competition any time soon, is she? But my guess is, she again is just trying to be nice!
So, let it all go over your head, count to 10 and it will all settle down soon, hopefully.
Oh! And keep your hair out of her reach. That is a bit cringe, I must admit.

Your problem is the personality of the woman, who just happens to be your MiL rather than specifically a general MiL problem, so unless one of us has ever come across someone who behaves in this way, it is difficult to know how to advise you, although she does sound intensely irritating and like you, I would find an over-tactile woman, would make my skin crawl. Can your DH not offer any advice, she is his mother?

Do you reply to every text and mail? If you do, why not cut that back and only acknowledge them every hour, gradually getting less frequent. If you have a young baby, they are very demanding of your time and you cannot respond every time.

Have you thought how would you deal with someone like this if you had to work with them? It does help one to look at the situation like this in a more impersonal way.

Apart from that all I can do is sympathise

Oh dear indeed. I'm amazed that mums with babies/young children have the time to write these lengthy posts.

Sorry Nella I don't have advice for you. For all I know, your MIL might be mostly a nice person with a few idiosyncrasies.

I blame social media.

What does your DH think about her behaviour?

Also, don't feel that you have to be constantly available or constantly replying to her texts. If she complains just say you're busy and choose a frequency - say once a week or something - that you can make time to chat.

I'd suggest deciding in advance how often you plan to see her. Once a month should really be sufficient. If she wants more you can just say that doesn't work for you. (It doesn't.)

If she tries to touch you just say you don't want to be touched. If she persists, move away from her or leave.

Don't tell her how much time you spend with your own mum.

Address her inappropriate behaviour in the moment. "What do you mean by that? I'm sorry but a visit every week doesn't work for us. Now how about that weather we're having? Never mind how much time he spends with my mum, what have you been up to?"

She doesn't sound toxic, just a bit overbearing. Best way to nip it in the bud I would say is see her on a schedule you can manage, keep to it, and don't let her complain.

Oh dear.

Perhaps someone will be along in a moment with advice, but I'm sorry, I don't have any.
Perhaps you could buy her a dog for Christmas - but, don't forget, a dog is for life, not just for Christmas.