If the alternator drive belt is tight then the alternator is dying.
First make sure all connections are tight.
Check the earth cable from battery to gearbox. These corrode inside the cover and cause poor connections, which could give your symptoms.
With the alternator off, it may be possible to replace the brush box/regulator assembly. As long as the slip rings inside are in good condition, a replacement regulator/brush box can cure the problem. Often the copper slip rings, that the carbon brushes rub against, will wear until their internal connection is exposed, then this tears the brush very quickly. If yours is like this, a replacement alternator is needed, unless a specialist near you can fit new slip rings.