and depending on your cast on, a larger needle may or may not be effective.
it helps a VERY LITTLE with a STANDARD LONG TAIL.
Stardard Long Tail (SLT)has 2 parts.
the THUMB STITCH and the INDEX stitch (that goes on the needle)
when doing a SLT, you 'snug up' the thumb yarn and reposition yarn with each stitch.
If you pull the thumb yarn tight, you'll have tight, non stretchy cast on no matter what size needles you make the index finger stitch on (and could end up with a ...../ sort of shape at the end of your knitting.)
A larger needle might help with a Knit, a Cable, a Crocheted, and other Cast ONs.
the REAL SOLUTION is to think about what you are doing as you are doing it, and NOT to tighten your stitches.
it take time, it takes practice, its a LEARNED SKILL.
we don't expect elemary school kids to write books.. but we do expect them to learn to write short letters, short essays, and (very) short stories.
High School kids are expect to write short (1 to 4 pages) reports..
its only in College/university that we expect real "PAPERS" (detailed, documented multipage reports on a topic)
(in other words we expect kids to learn to read and write age 7, but we expect 10 or more years of practice before thay are good at it!)
cut yourself some slack, and PRACTICE knitting.. (looser, and looser cast ons!) don't expect perfection in a month.. it take a few year (ok, so not 10 maybe (or maybe 10 and more!) but just because a 7 year old can read and write, they are not writers.. and a knitter of 1 year can knit and purl, but they often still need a lot of practice!
other alternates are VARIATIONS of SLT--like Norwegian Long Tail (aka german/german twist/twist) cast on.
the NLT has a twist in the thumb yarn that adds extra yarn. (you also need a long tail of yarn to start with NLT (vs SLT) the twist adds extra yarn to 'thumb stitch' and this extra yarn adds extra 'stretch' (even if it is work tight)
and NLT is just one of MANY long tail 'versions' (i know at least 8 version of LT alone!)