Save the money on the handyman.
The good news is that your toilet uses standard parts. And it's easy to do what you need to do here.
I think it would be a worthwhile project to replace the fill valve and flapper in your toilet.
However, before you do that, try adjusting the "chain" length of the current very-worn flapper. You have way too much slack in the chain at present, and the amount of slack affects how the flapper operates. See my comments below. Of course, the flapper looks so worn it's hard to believe that it makes a seal.
Here's a video from the manufacturer named Korky on how to replace your old fill valve with a Korky 528 quietfill MaxPerformance valve. That's a valve we recommend.
Here's how to install a flapper. I think you want a Korky 2001BP flapper -- i.e. the standard one. The video says to install the chain "with minimal slack". This means that you adjust the chain length so there is going to be one, but only one, link of chain resting on top of the flapper when the handle is at rest. There are lots of links of chain on a Korky flapper, and you just put the clip on the chain where you need to in order to achieve that. If the thing doesn't work properly, try one link more or one link less, and make sure the lever you attached it to is directly above the flapper. A simple adjustment of chain length by one or two links in either direction makes a huge difference. Surprising, but true.
I am actually surprised that your toilet flushes at all with the sloppy amount of slack in your current "chain".