getting from Paris St. Lazare train station to Paris Est

Too far to walk, especially with bags. There are various Metro routes but all involve changing trains, which can take awhile in Paris stations. I'd spring for a taxi. There will be a taxi stand just outside St-Lazare. Here and here are lots of info about the station. You can also find Gare de l'Est on those websites.

With the Metro, take the No 3 line from St Lazare to Reaumur-Sevastopol and switch to the No 4 line to Est, taking 16 minutes including walking time. Or take the No 27 bus, then a 400 yd walk, taking about 20 minutes including walking time.
https://www.ratp.fr/en/itineraires#no-back

I've done this. I took a taxi. It took about 15 minutes with traffic. Cost about 10 Euros.

In June 2022, we (parents and young teens) decided to walk between these stations with our backpacks and roller bags. It took longer than Google predicted (maybe twice as long with the luggage and we had to consult the map a few times) and we were quite hot by the time we got there. As we approached Gare de l'Est, a father with a younger child came up to us and asked if we would like popsicles. He had purchased a box and he and his daughter couldn't eat them all before they melted. Despite the popsicles, I would probably not walk if I was doing this again.

Thank you all for your input ! This really helps.

my choice might well be RER-E one stop from Haussmann Saint Lazare to Magenta (near Gare du Nord) and then walk, either down the staircase or around on the streets.

Many ways to skin the cat.