I've always understood that a chain & cassette should have the same level of wear as one another & be replaced at the same time - so a new chain performs better with a new cassette (& 8-spd Shimano spaced cassettes are dead cheap which is good)...
Two more possible reasons though is that (a) the bottom bracket shell area could actually be flexing when your pedalling hard - e.g. uphill - or 'under load' as you say - which means that your are physically moving the gear cable a little where it runs under the bb shell & carries on to the rear derailleur & that's causing the chain to try & skip to the next gear, or (b) another reason is that the cables themselves are old or rusty preventing the gear indexing from engaging correctly.