Sounds like you could be a candidate for either losing pressure or excessive pressure. The compressor is not cycling because your pressure is either too high or not high enough. If it's OK after sitting all night, then I would bet on high pressure. Once the system loses enough pressure to hit the low pressure cutoff, it usually just stays off. Have you done any recharging recently. The split in the hose might be a good place to start sniffing for R-12 leaks. Start by having your pressures checked and yes replace that low pressure hose before it ruptures and you have a bigger problem to deal with. BTW, when you say it works for a while, how long is a "while"??
2000 Dodge Grand Caravan SE