Would be glad to

I was writing an email and faced the following situation:

I am coming to Oxford this summer. If you are still around, I would be glad to hang out with you.

Now normally we don't construct the conditional statement like the one above, where a present verb is followed by would. But if I change the second one to will it kind of becomes very direct and that's not what I want. So is the construction above still correct or should I go with something else? If it's correct, could someone explian the technical details behind it?

This is perfectly grammatical. You can use a present-tense verb with "if" to talk about the possibility of a future event - check out my answer on Future tense in conditional clauses for more details.

For the second clause, your intuition is correct; "would" and "will" mean the same in this context, but "would" is a little less direct.

Firstly, I think in conversational language the rule around having to use past tense in the conditional clause when the main clause is in subjunctive are often not followed, so it's fine.

If your main concern is not wanting to sound too direct, I would rephrase the sentence to: "If you are still around then, it would be great to hang out."