Two pushsticks is the way to go, and RK at the least.
You can't concentrate on everything at the same time.
I remember the first time I touched the blade with a piece of hardwood, unintentionally I might add, as I had that sorta thinking that one was sufficient enough, and would make do with that until I found some soft ply or MDF.
Quite the bang when the blade snatched the timber into the ZCI!
Could be a lot of issues here, with new blade and combination square, use the pick a tooth method and line up with mitre slots.
Short fence made and support for outfeed presuming you would sort out.
With that sorted
Is there a dish on the table,
Is the wood surfaced face and edge
Is there runout on either the arbor, bearings possibly,
although I'd want to also check flanges very carefully as those can make a lot of wobble should there be grit or damaged, most meant to have contact with blade on perimeter.
Just a few possible reasons,
Mastering your tablesaw, by Kelly Mehler is one of the best videos I've seen available on youtube, although dated and American, exceptions can be made for the setup of the setup of the rip fence having toe in, as I guess there might not have been enough time or whatever in those days.
Gwinnet woodworks have two video seminars on safety which are pretty thorough.
the older even more so.
Steve Maskery has made some good articles also.
Just for some tips and whatnot.
Keep safe
Tom