If your battery is low it'll cause the car to behave this way too because the alternator is having to do all the work and struggling with the battery. Check the fluid levels in the battery if it's a not a maintenance-free one and either way put it on charge for at least 12 hours. If you alternator is on it's way out you should get the same symptoms after about 6-8hrs of driving time(about a weeks commute ish).
Another much quicker way is to put an ampmeter on the battery terminals with the engine running. If the alternator is good you'll get a reading of about 13.5v, anything lower than 13v and it's time for a new alternator. If you go to a local garage or mot station they'll usually do this for free because it's a 2 second job.
As for replacing your battery you have 2 options, 1st is,as you say get a higher ampage battery or a battery designed for a diesel engine, 2nd option, is a lot more work but run a split charge system and run two batteries, 1 for ICE and 1 for rest of the car. Split charge is easy and cheap to setup but you have to find somewhere to house a 2nd battery. It depends on what audio setup you have as to whether you really want it. That bit is your choice
