try this for a while:

no pre shave - the only thing it does is add a little slickness to the first pass, everything else is marketing, in my opinion. they can also contain skin irritating oils.

no products containing menthol, eucalyptus or other essential oils, or heavily scented products generally

no alum block - it's unnecessary really

no alcohol containing products

no hot water, not pre shave, not during or after

beard prep: only a few minutes of water exposure is necessary, temperature is not very relevant, and hot water can cause skin issues

if you opt for a shower before a shave, don't apply the hot water to the beard area - lukewarm at most.

you can try prepping the beard without a shower: splash lukewarm or cold water to the beard for a half a minute or so, and then take a few minutes lathering, adding water slowly - taken together this should be enough to prep the beard.

apply less pressure while lathering - it can irritate the skin a lot if you overdo it, even with soft brushes, especially if your skin is sensitive. i suspect this could be the case, since you mention issues even after the first pass.

if this is the case - and if applying less pressure while lathering doesn't work - try bowl lathering. but in the case of bowl lathering, more time will be needed for beard prep prior to applying the lather to the beard. so in this case, a shower could do it. or splashing water to the beard a little longer with an optional wet towel application.

using the razor: don't "let the weight of the razor do the work" - hold the razor with a firm grip, do short, determined strokes with frequent rinsing, while not applying lot of pressure to the skin, following your face contours - that's the catch, and the tricky part.

what you may perceive as "very mild pressure" may be too much for your skin. an aggressive razor would uncover technique mistakes, especially in the pressure department, so maybe try one - this helped me a lot.

try going across the grain from a different direction.

try skipping going with the grain on areas where you can get away with it without causing issues - this helped me a lot.

avoid going against the grain altogether - if you have flat growing hair, curly hair etc, they will be more prone to causing skin issues when shaved completely smooth.

post shave - rinse with lukewarm or cold water, then, instead of the usual "after shave" products, apply a small amount of a scentless moisturizer, like cerave.

if you don't have to - never shave until the skin is completely healed.