Yes, cat5e exceeds cat5 so buy cat5e.
Cat6 exceeds cat5e, so if that's also $5 or not too expensive then get that instead.
you are using 56k dialup right? if so, then a modem being the problem is not uncommon. Check to see if there are updated drivers for your modem also. Depending on the computer and it's age it may be easier to just buy a new modem, ebay stores should have them for $20.
with dsl for other reasons, the common & best thing to do is put a dsl filter at the NID box outside the house then run a direct line from the dsl side of that to your modem for best possible connection. Your outside wire is thicker because of it's insulation, if it is phone wire.
Running a dedicated phone line from the box outside directly to your computer would bypass any potential problem so as long as your ok with doing that, it would be fine and should tell you where your problem most likely is. If you use cat5e network cable as opposed to cat3 phone line, there will be 4 pairs I think of wire- you only need to use 1 pair (2 wires). Doesn't matter which pair but it's usually blue/blue-white. Be aware that if this cat5/e cable you buy has connectors on it, they are rj-45 connectors which are not phone line connectors. Phone line connectors, the one that plugs into your modem or phone, are rj-11. With cat5/e cable you will have to put an rj-11 plug on one end of it, or connect it to a wall jack with screw terminals and then use regular phone line from that to your computer modem.
