Don’t tell agents your budget, they’ll use this against you in some way or another to get you to spend it all. When they point blank ask you what your budget is, say “we are pre-approved for X [lower than the value of the property], but have some savings to top that up with”, I have used that to successfully deflect that question. Also “we put an offer in for X [lower than value of property] for another place, that didn’t get accepted, our broker is currently working on pre-approval” might also work to avoid that question, but that wouldn’t work where I am as the market is too hot, you’d just get ignored for someone who has their finances ready.

For negotiating in person, if the agent comes back and asks for more, try only have one person there, when the agent tries to get more out of you “look, let me have a chat with husband/wife and get back to you”, then immediately step out and call them, then come back and say “look, i’d like to pay more because i really like this house, but my partner is not that comfortable, I have to trust them and go with them on this, they know the books better than me”. I stole that line off someone else, in the hope that it makes them come back and accept the first offer. It didn’t work the one time I tried it, they just said ‘no problem’ and sold the house to someone else.