8 way splitter degrades internet signal over time
It sounds like you might just be dealing with a bad splitter, so my first recommendation would be to just replace the splitter.

There is a loss associated with every port on the splitter.  If you look on the splitter, you'll probably see something like -4dB printed on each port.  This is how much signal loss there is on that port.

Typically your cable modem will be connected to a splitter directly from the source feed, then all other TVs split from a secondary splitter.

Amplifiers usually just add noise and if there is a signal quality issue, it's best to work with your service provider to get the addressed.

Bad connector fittings on the end of the cable can make a huge difference, as can the length and quality of the cable used. By SOP, most cable installers would probably replace every RG6 compression fitting between the source and the modem.