But also means however.

Both in the Oxford and Cambridge dictionaries, I found the word "but" used in comparable contexts (meaning however), as in the examples provided in the OP.

however = used for adding a comment to what you have just said) although something is true.

1. Oxford dictionary but meaning https://g.co/kgs/JtUk57g however; and yet

I’d love to come but I can’t make it till 8 o’clock.

Cambridge dictionary https://dictionary.cambridge.org/grammar/british-grammar/but

But as a linking word We use but to link items which are the same grammatical type (coordinating conjunction). But is used to connect ideas that contrast.

They’ve bought a house in Manchester, but they still haven’t sold their house in London.

'However' has been used in place of 'but' in each of the five sentences to make the meaning more clear.

[1] She lost a great deal of respect for him in that moment, but she would never admit it.

She lost a great deal of respect for him in that moment, however she would never admit it.

[2] He stared at the mess, but he tried to hide his unease.

He stared at the mess, however he tried to hide his unease.

[3] He went the pub, but she wasn't there.

He went the pub, however she wasn't there.

[4] He started crying, but he pretended that he wasn't.

He started crying, however he pretended that he wasn't.

[5] Maybe I'm wrong, but correct me if I am.

Maybe I'm wrong, however correct me if I am.

It's generally understood that the coordination conjunction 'but' is used to create contrast between different constituents (e.g. positive against negative, good against better, bad against worse)

Some of the sentences may not sound proper with 'however' but the Idea is to convey that 'but' doesn't have only one meaning but can be replaced with other words like yet, nevertheless, nonetheless, even so, however, still

But also means however.

Both in the Oxford and Cambridge dictionaries, I found the word "but" used in comparable contexts (meaning however), as in the examples provided in the OP.

however = used for adding a comment to what you have just said) although something is true.

1. Oxford dictionary but meaning https://g.co/kgs/JtUk57g however; and yet

I’d love to come but I can’t make it till 8 o’clock.

2. Cambridge dictionary https://dictionary.cambridge.org/grammar/british-grammar/but

But as a linking word We use but to link items which are the same grammatical type (coordinating conjunction). But is used to connect ideas that contrast.

They’ve bought a house in Manchester, but they still haven’t sold their house in London.

'However' has been used in place of 'but' in each of the five sentences to make the meaning more clear.

[1] She lost a great deal of respect for him in that moment, but she would never admit it.

She lost a great deal of respect for him in that moment, however she would never admit it.

[2] He stared at the mess, but he tried to hide his unease.

He stared at the mess, however he tried to hide his unease.

[3] He went the pub, but she wasn't there.

He went the pub, however she wasn't there.

[4] He started crying, but he pretended that he wasn't.

He started crying, however he pretended that he wasn't.

[5] Maybe I'm wrong, but correct me if I am.

Maybe I'm wrong, however correct me if I am.

It's generally understood that the coordination conjunction 'but' is used to create contrast between different constituents (e.g. positive against negative, good against better, bad against worse)

Some of the sentences may not sound proper with 'however' but the Idea is to convey that 'but' doesn't have only one meaning but can be replaced with other words like yet, nevertheless, nonetheless, even so, however, still (Ref. Oxford dictionary)