Learn French: Je lâche des mailles, mais je continue tout de même. - I drop stitches, but I continue all the same.

They mean exactly the same thing ("all the same" or "anyway") and are fully interchangeable here. Tout de même is simply a bit more formal or elegant, whereas quand même is extremely common in everyday spoken French.

In English we could say "but I still continue". Can I put tout de même before the verb to match this?

No, French is very strict about not separating the subject pronoun (je) and the verb (continue). Concession connectors like tout de même or quand même must be placed after the conjugated verb, never before it.

Why use lâche for "drop" instead of laisse tomber or fais tomber?

The verb lâcher means to "let go of" or "release". In knitting, you accidentally "let go" of a stitch from the needle, so lâcher une maille is the standard term for dropping a stitch. Faire tomber is used when you physically drop an object onto the floor (like a glass or your keys).