Wheel sizing is kooky.
For whatever reason 700c wheels are (nowadays)(sometimes) called 29rs which is smaller than 27".
https://www.sheldonbrown.com/tire-sizing.html
IMO- I used to advocate staying with 27" as there's good tires out there- namely the Paselas and the Sand Canyons. (there's also the Continentals that I haven't used). But, really, it's just a handful- tires REALLY change the complexion of your ride- and it's a shame to limit your ride because of an obsolete wheel size.
You've got a while before spring, do a ****-ton of reading- watch some RJ The Bike Guy videos on the YouTubes. Look out for the problems that people have in upgrading and swapping parts.
IMO- you NEED new wheels- you're running steel rims. Not only is that going to significantly increase the weight of your bike, but most brakes have difficulty grabbing on steel rims (hence the 'grippy' tread on the rims, which helps a
little.) Looking at your pix, you definitely have enough room on your front brake to go down the 2mm. You should have enough on the rear as well- but more experienced people should be able to give you a better answer.
IME- if you decide to get used 27" wheels off eBay for a good price- don't forget to factor in the cost of shipping. Check Craigslist of FB for some used 27" wheels- a lot of people have a lot of extra 27" wheels after changing to 700c. ONLY get aluminum rims.
Again IMO- don't go throwing a whole lot of money at this bike- you have an excellent 'get around' bike, but you'll never turn it into a racing machine. New/different wheels, new brake pads, new tires, new cables/housing and fresh grease all around... you'll have a really slick, fun bike.