Your mountain bike would drive me nuts for commuting around town. Based on your description of the condition, my bet is that the suspension is also shot (dried out and/or worn o-rings, busted valves). Most suspension components need servicing every 30-100 hrs of riding (probably a little less often if you're only on street).
Buy a used touring bike, in reasonable condition: chainrings, cassette, bottom bracket, headset in good condition. Get a rack with saddlebags to get the weight off your back, and fenders to keep off the spray on wet days. I also second getting tires that are a little wider than the typical road bike. Also recommend a little bit of tire sealant in the tubs - really helps prevent flats.
RE maintenance: Every bike needs regular maintenance. You can buy new to avoid it for a while, this buys you maybe a year if you're commuting daily. However, you aren't going to buy a new bike every year (I hope!) so you may as well learn to do it yourself. You can do almost all maintenance with a few tools and a little know-how. Tools: chain whip, cassette tool (specific to your cassette type), crescent wrench, allen key set, chain break tool, crank puller (for your specific cranks), valve stem tool, chain wear indicator, bike tire levers. These tools are cheap, small/compact, and you only buy them once (can get most in a kit for about $100 -
https://www.amazon.com/dp/B017KS56MG/). It's simple to replace the chain which should be done at 0.75% wear to extend the life of your chainrings/cassette. And the chainring(s) and cassette are also easy and cheap to replace on your own. Not sure about EU prices, but I get new KMC chains for about $10, 8-speed cassette for $17, and chainrings for about $20 each.
If you're serious about bike commuting for the long-term than doing your own maintenance will save you a ton of time and money, and you can be sure the work is done right.