I'm a beginner at fixing bikes, but usually I'm able to find an answer in this forum. Not this time however.
When I ride on any tiles, the wheels tend to follow the edges between the tiles. Even the slim grooves between regular tiles you find on a sidewalk.
It feels very unstable, to the point of feeling insecure when riding.

My bicycle has alesa rims, marked 622* 700c. Its got 32-622 tires on it, the inner width of the rim is 21mm. The QR axle isn't bent, it's also as tight on the frame as I can get it.
The only issue I can tell is the fact that the tires aren't fully seated in some places on the rims.
I am kinda low on money right now, so I'd like to not spend it at the lbs if possible.

Don't ride in the grooves in pavement. Absolutely normal for a tire to want to "follow" the groove. If you had a tire as wide as on a car, you might not notice it.

Are you keeping enough air in the tires? Usually if the bike feels squirrely from the tires it's a low tire pressure. However tires do tend to follow grooves and cracks. Not sure what you mean about the tires not being fully seated on the rim. Sometimes you can feel a tire being out of round from such. Is riding on the sidewalk legal where you are at?

I understand but I've never had this issue on any other bicycle.

I mean the tire isn't as deep in the rim in certain spots. And no I'm not on the pavement, some parts of the city have tiling for the bike lanes

You skipped past the question in the first line of the post that you responded to. Check your tire pressure.

The answer changes a bit if you are riding super cheap tires.

try riding on a grid bridge deck some time

Add rain, and now we're talking. Then there's the vertigo I get from looking at the water 50' below!

Try to avoid crossing the grooves at close angles and when you do have to cross a groove at a close angle, pop your front wheel up just a little so the front wheel won't get caught in there.

What were your previous bikes and did they have the larger diameter 700C tires on them? What model of bike is this one? Perhaps what you are feeling as insecure is just getting comfortable with a larger wheel and larger performance oriented bike frame that might actually be feeding you the feeling of the road more.

However do check your tire pressure often. If it's been a week since you last checked, then it's well past time to check. On my narrower 25mm tires I check them and put air in them every 2nd or 3rd day. The other thing to check is whether all the spokes feel about the same tension to your hand/fingers and they aren't too loose altogether.

Don't know about your area, but bike shops here are pretty much owned or managed by the mechanic. And they all are willing to look at your bike and give you their assessment of what it needs for free. Whether you leave the bike with them or take it home to do the fix yourself is okay with them.

Try to shift your weight back and let the front tire float over the uneven surface. Don't grip the bars tightly, pretend they're raw eggs you don't want to crush (OK probably not quite that loose, but...)

Here are some tips for riding on cobbles that might help.

Dirt Riding 101 class time..
when riding in rutty terrain, it's best to DECOUPLE YOUR WEIGHT from the bike...you can do this by STANDING on the pedals, and removing your butt from the seat... Also, make sure that your thighs are NOT touching the frame's Top bar, or seat.
this allows the bike to shift around into different ruts, but your mass doesn't cause the bike to shift more than needed.(a weight in motion wishes to remain in motion)
and quit death-gripping the handlebars.

and i hate ruts/grooves almost as much as sticks/roots crossing trails at an angle...
and then... there's those wet railroad/trolley tracks.... bunny-hop them suckers, or carry lots of band aids.

That's a normal phenomenon known as tramlining, most road bikes do it to some degree, wider tyres seem less susceptible. My first concern if your bike feels really bad would be that the headset or front hub is worn or improperly adjusted, or the frame and/or fork might be bent or cracked. If none of those, you could try changing your tyre pressure - higher pressure may feel more sensitive to road defects but react more directly to your steering input. Also shifting your riding position to put more or less weight on the front wheel can make a difference.

The OP, aTechnical, stated this in the first post which means that the tires need to be properly seated before moving on to anything else. If these are tires with tubes try letting all the air out of the tire and then push the bead in checking for any tube caught under the bead. Push the tube back into the tire and inflate which should set the beads in place.

Bike HEADSET Care...
does your bike have a threaded headset?
If the Steering tends to follow "lines and grooves", and feels stiff or notchy, then the Headset is either way too tight or the grease has failed... the first option is fixed by SLIGHTLY loosening then re-tightening the Top cup and Top Nut of the headset, and ensuring that the steering now works smoothly and that there is no "Clunky" looseness of the headset... the Second option is a complete servicing the headset by disassembling , cleaning, inspecting , CAREFULLY AND THOROUGHLY, for Any Wear or rust pitting, then replacing the assembly if rusted or pitted...

the second option should be done only by a competent mechanic . Both of these "Threaded Headset" operations are shown in videos on YouTube...

if the bike has the more modern "Non-Threaded" Steering stem and Straight steerer tube/fork... the tension applied by the Top Cap/screw sets the preload of the steering bearings in the Headset, and will require that a specific procedure be followed... check for drag/roughness/ clunky operation of the steering... Problem d=found? loosen top cap, loosen steering stem clamp, SLIGHTLY tighten top cap screw until no "clunky" is felt, but no drag is present... Tighten Stem clamp screws to proper torque(OVER-Tightening will cause possible breaking of the stem, and possibly your face, too!) then i loosen/re-torque the top cap screw.

you can also find this procedure on youtube.

if any of these procedures cause you to worry... take the bike to a shop for repair, ok?

i hate seeing anyone getting hurt.

oh, and there MAY be a bad Frame bend or break causing the erratic handling too.
no one has mentioned THAT, from what i've seen.
I'd seriously doubt it's got anything to do with the tires themselves ..
other things to consider... poorly centered wheels... a bent Fork... badly cut Dropouts causing wheel misalignment(found on New WallyMart-quality bikes, all too often)... badly Installed wheel(s).... and, another WallyMart thing, the Backwards Suspension fork (fixed one of those, two weeks ago)

59th Street bridge in NYC in the rain was interesting. Nothing like road rash that looks like Belgian waffles, ouch!...