I've been doing pull offs until now by anchoring the lower fret and sliding my finger down off the higher one. The problem is this doesn't work for doing multiple pull offs down consecutive strings. I tried to simultaneously pull off and hammer on but I keep bending the string before the pull off and it sounds strange to me. Any tips to help me do it right?

"sliding my finger down" What do you mean by this? A pull-off is lifting a finger such that the higher note is immediately followed by the lower note, without picking the string.
"bending the string" Pull-off technique can involve a little flick to make the lower note sound louder or more percussive, but it shouldn't actually bend the string so much that a sharpened pitch is heard.
Do you have any examples?
Sliding down towards the anchor finger sounds like it's not the right technique. Pull offs will have you kinda pluck the string with your fretted finger.

If you're trying to do it across multiple strings, you can try barring the "anchor"/lower note if they're both on the same fret.
If they're on different frets, you can apply a similar principle and flatten your index finger to span across a few strings and hover (lightly touching the strings for muting purposes) into place.
This will help mute the other string so it doesn't ring out from totally abandoning the one you left, and you get a cleaner melodic line.

Hope that helps!
"Every time you pick up your guitar to play, play as if it's the last time" -Eric Clapton