Learn English – Like a Swiss watch

Mark my words, the phrase isn't all that outdated.

Nevertheless, I tried to think of other alternatives to this phrase, since that's what you asked for. I manage to think of a few:

• Don't say I didn't tell you (along with other variants)
• ...you can bank on it
• You can bet your bottom dollar
• I'm telling you...
• ...I guarantee it

Here are some excerpts where these phrases mean roughly the same thing as mark my words:

Don’t say I didn’t tell you: Two years from now, the GOP will officially split into two parties... (from a news article by Charles Ellison, 2014)

And you can bank on this: while crusades may start out as one-man crusades, if the idea behind the enterprise is good, soon you'll have lots of support. (from Magic of Thinking Big, David Schwartz, 1987)

We can teach all the right responses in the world but if we never role model them - well, you can bet your bottom dollar you won't see them in your children. (from a book on parenting by Eydie Comeaux, 2003)

I'm telling you, there's only one way you gonna get to Norlins now, and that's by cab. (from Old Glory: A Voyage Down the Mississippi by Jonathan Raban, 2011)

You can replace the bolded words with mark my words, and the passages will pretty much mean the same thing.

As for the currency of mark my words, an Ngram hints that its usage may have peaked about 100 years ago. Yet even in recent years it still dwarfs some of the alternatives I've mentioned:

I think the most interesting part of that Ngram, though, is the sudden spike in I guarantee it, which seems to coincide with Joe Namath's famous Super Bowl prediction, further discussed in this column.

I often give in to her.

is indeed a correct sentence, although you might want to consider:

I often defer to her.

They both mean that you are letting the other person have their way. However, give in might be construed as weakness, or as a way to end a protracted argument, whereas defer might be interpreted more as being polite and gentlemanly.