The key to getting your offer accepted is to not get married to a house you don't own. Your offer WILL be accepted, on a certain house, by a certain seller. Your negotiating position is exactly as strong as your ability to walk away from it, vs the sellers'. If you decide that for some reason this house is the only house you could possibly ever buy, the sellers can set the highest price you could possibly pay. If the sellers need to leave the country within the month, you can set the price much lower.
It's important to be able to weigh up the relative strengths of your positions in a negotiation, but I'd say as a frame challenge that it's more important to put yourself in a situation in which you can't lose.
I was in almost your exact position last October. The only difference was that I had a 20% "deposit" (as in 80% LTV, which only matters to the bank, not the seller). I considered myself as having a generally strong position. Not only was I a first-time buyer, I also had an incredibly large catchment area, and very loose requirements - only really caring for square footage.
I made an offer on a house, it was accepted, I payed fees to secure the mortgage, perform searches and surveys, and assumed I'd have to pay solicitors' fees. The solicitors discovered that the house was not freehold, as had been advertised, and this was a dealbreaker for me. I liaised with the sellers, and reiterated my position - I would like to purchase the house, as it was advertised to me, for the price upon which we agreed. The seller's position was that they wanted me to cover the cost of buying the freehold, and that they wanted to buy a specific house that was to be sold to them off the market, to house their growing family.
My ability to walk away from that negotiation was greater than theirs. I cost myself no extra money, and my situation remains the same. I can continue searching for a house. I didn't budge on my offer, expecting the sellers to acquiesce.
Their house remains for sale, and I lost no further money in the endeavour. If they had reaccepted my offer, I would have closed and moved in. Either outcome was fine for me.
I made an offer on another house of £15k less than the listed "offers in region of" price, (about 10%). The estate agents took this to the sellers, who tried to get a higher offer. My position was as with the previous house, and the sellers' position was liquidating their late parents' house. I imagined their reasons for doing so were for simple monetary gain, and that they were in no rush. Both the sellers and I were able to painlessly walk away from the deal, so I said that I'd leave my offer open and continue looking. As a nice surprise, they accepted the offer, so here I sit, typing this answer. Similar to the previous situation, either outcome was fine for me.
As you can see, whether my offer was accepted or not had little bearing on the more important factor - how good my life is.
Make sure to familiarise yourself with the process of buying a house, and have enough money set aside to cover the expenses. Assume that any costs to do with investigating the house and conveyancing will be lost on every offer, as these are generally things you put yourself on the hook to pay before the seller is contractually obliged to go through with the sale. Make preparations to have these processes completed as quickly as possible to give the sellers less time to pull out between accepting your offer and exchanging contracts. I found that my solicitors were by far the slowest part of the process, so it could pay to pay a little more for good ones, or to do the conveyancing yourself (assuming you have the time).
It's pretty standard to request the seller remove the property from the market upon acceptance of your offer, and all of the estate agents with whom I interacted said it was standard procedure to cancel all viewings for a house upon which an offer has been accepted, (and not book any further viewings).
It's also worth trying to get a measure of the sellers' characters, if possible. If they seem like someone who's chancing it, and will play you around, save yourself the hassle and the money, and walk. Estate agents generally prefer you don't have any contact with the sellers, so this isn't always something you can do. The estate agents are being paid by the seller when the house is sold, so you can't believe anything they say either.